L(s) = 1 | − 32·2-s + 640·4-s + 278·5-s + 548·7-s − 1.02e4·8-s − 1.76e3·9-s − 8.89e3·10-s + 7.39e3·11-s − 1.75e4·14-s + 1.43e5·16-s − 2.83e4·17-s + 5.65e4·18-s + 9.98e4·19-s + 1.77e5·20-s − 2.36e5·22-s + 3.33e4·23-s − 7.41e4·25-s + 3.54e4·27-s + 3.50e5·28-s − 9.31e4·29-s + 3.11e5·31-s − 1.83e6·32-s + 9.06e5·34-s + 1.52e5·35-s − 1.13e6·36-s + 9.63e3·37-s − 3.19e6·38-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 5·4-s + 0.994·5-s + 0.603·7-s − 7.07·8-s − 0.807·9-s − 2.81·10-s + 1.67·11-s − 1.70·14-s + 35/4·16-s − 1.39·17-s + 2.28·18-s + 3.34·19-s + 4.97·20-s − 4.73·22-s + 0.572·23-s − 0.949·25-s + 0.346·27-s + 3.01·28-s − 0.709·29-s + 1.87·31-s − 9.89·32-s + 3.95·34-s + 0.600·35-s − 4.03·36-s + 0.0312·37-s − 9.44·38-s + ⋯ |
Λ(s)=(=((24⋅138)s/2ΓC(s)4L(s)Λ(8−s)
Λ(s)=(=((24⋅138)s/2ΓC(s+7/2)4L(s)Λ(1−s)
Degree: |
8 |
Conductor: |
24⋅138
|
Sign: |
1
|
Analytic conductor: |
1.24287×108 |
Root analytic conductor: |
10.2755 |
Motivic weight: |
7 |
Rational: |
yes |
Arithmetic: |
yes |
Character: |
Trivial
|
Primitive: |
no
|
Self-dual: |
yes
|
Analytic rank: |
0
|
Selberg data: |
(8, 24⋅138, ( :7/2,7/2,7/2,7/2), 1)
|
Particular Values
L(4) |
≈ |
2.978132339 |
L(21) |
≈ |
2.978132339 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Gal(Fp) | Fp(T) |
---|
bad | 2 | C1 | (1+p3T)4 |
| 13 | | 1 |
good | 3 | C2≀S4 | 1+589pT2−1312p3T3+198200p3T4−1312p10T5+589p15T6+p28T8 |
| 5 | C2≀S4 | 1−278T+151453T2−10092158pT3+507231524p2T4−10092158p8T5+151453p14T6−278p21T7+p28T8 |
| 7 | C2≀S4 | 1−548T+205477pT2−1005153868T3+1410523587416T4−1005153868p7T5+205477p15T6−548p21T7+p28T8 |
| 11 | C2≀S4 | 1−672pT+69657359T2−362239950472T3+1972371186671352T4−362239950472p7T5+69657359p14T6−672p22T7+p28T8 |
| 17 | C2≀S4 | 1+28316T+80288674pT2+22995691781952T3+723743706605364939T4+22995691781952p7T5+80288674p15T6+28316p21T7+p28T8 |
| 19 | C2≀S4 | 1−99888T+6302386479T2−273388952905776T3+9264012062067280120T4−273388952905776p7T5+6302386479p14T6−99888p21T7+p28T8 |
| 23 | C2≀S4 | 1−33388T+4497107523T2−11678709274332pT3+15457403175217183288T4−11678709274332p8T5+4497107523p14T6−33388p21T7+p28T8 |
| 29 | C2≀S4 | 1+93140T+34300522218T2+2719238895824544T3+84⋯75T4+2719238895824544p7T5+34300522218p14T6+93140p21T7+p28T8 |
| 31 | C2≀S4 | 1−311160T+140644653260T2−26779795197492120T3+62⋯06T4−26779795197492120p7T5+140644653260p14T6−311160p21T7+p28T8 |
| 37 | C2≀S4 | 1−9636T+9745737498pT2−1878062788215984T3+50⋯59T4−1878062788215984p7T5+9745737498p15T6−9636p21T7+p28T8 |
| 41 | C2≀S4 | 1+82892T+552926747522T2+95958507863966880T3+13⋯27T4+95958507863966880p7T5+552926747522p14T6+82892p21T7+p28T8 |
| 43 | C2≀S4 | 1−569264T+675637103839T2−373504087629850528T3+21⋯84T4−373504087629850528p7T5+675637103839p14T6−569264p21T7+p28T8 |
| 47 | C2≀S4 | 1+574200T+1181775847100T2+840326241178871192T3+77⋯90T4+840326241178871192p7T5+1181775847100p14T6+574200p21T7+p28T8 |
| 53 | C2≀S4 | 1−1235350T+2684236572461T2−3308983772395621830T3+45⋯08T4−3308983772395621830p7T5+2684236572461p14T6−1235350p21T7+p28T8 |
| 59 | C2≀S4 | 1+231504T+2410408632647T2+675371015279844664T3+89⋯72T4+675371015279844664p7T5+2410408632647p14T6+231504p21T7+p28T8 |
| 61 | C2≀S4 | 1+685684T−2963552678006T2+807290360276097248T3+21⋯11T4+807290360276097248p7T5−2963552678006p14T6+685684p21T7+p28T8 |
| 67 | C2≀S4 | 1+3271056T+25389643205903T2+59152713001018797288T3+23⋯16T4+59152713001018797288p7T5+25389643205903p14T6+3271056p21T7+p28T8 |
| 71 | C2≀S4 | 1−175012T+22437418387579T2−7492697774427458180T3+26⋯04T4−7492697774427458180p7T5+22437418387579p14T6−175012p21T7+p28T8 |
| 73 | C2≀S4 | 1−7137890T+48989573892505T2−21⋯78T3+81⋯12T4−21⋯78p7T5+48989573892505p14T6−7137890p21T7+p28T8 |
| 79 | C2≀S4 | 1+7053952T+797675870260pT2+36⋯40T3+17⋯66T4+36⋯40p7T5+797675870260p15T6+7053952p21T7+p28T8 |
| 83 | C2≀S4 | 1−657288T+707565190612pT2+83704316584548117240T3+16⋯26T4+83704316584548117240p7T5+707565190612p15T6−657288p21T7+p28T8 |
| 89 | C2≀S4 | 1−11452234T+85829145588041T2+87845546274720266406T3−15⋯60T4+87845546274720266406p7T5+85829145588041p14T6−11452234p21T7+p28T8 |
| 97 | C2≀S4 | 1−428002T+262966365990817T2−17⋯42T3+29⋯32T4−17⋯42p7T5+262966365990817p14T6−428002p21T7+p28T8 |
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L(s)=p∏ j=1∏8(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−7.03756303518920195876170155139, −7.01339613694661659856620335418, −6.70821487682950512251273605229, −6.56363104845749508210945193692, −6.46821681891209597350460649470, −5.79855245320963747002061246060, −5.70975673505344896551169212456, −5.57867484772998308280544509253, −5.37460375742095279129330887643, −4.88923701035266010361412830962, −4.44324204928085912764662509741, −4.19496608143432468790937241975, −3.92625149448016333001552643943, −3.20527214781971589569875194324, −3.16836092067020112941704425172, −2.87267500947223221718814603193, −2.74311444586108485849332207696, −2.05549331540689843672386537320, −1.97027879172249492254003579745, −1.60443714002249210641926510294, −1.47826896211398055483076364582, −1.11169766386835853901130513463, −0.77963620483203718354731703557, −0.56658744512281102407946892852, −0.33240100619318930520040079901,
0.33240100619318930520040079901, 0.56658744512281102407946892852, 0.77963620483203718354731703557, 1.11169766386835853901130513463, 1.47826896211398055483076364582, 1.60443714002249210641926510294, 1.97027879172249492254003579745, 2.05549331540689843672386537320, 2.74311444586108485849332207696, 2.87267500947223221718814603193, 3.16836092067020112941704425172, 3.20527214781971589569875194324, 3.92625149448016333001552643943, 4.19496608143432468790937241975, 4.44324204928085912764662509741, 4.88923701035266010361412830962, 5.37460375742095279129330887643, 5.57867484772998308280544509253, 5.70975673505344896551169212456, 5.79855245320963747002061246060, 6.46821681891209597350460649470, 6.56363104845749508210945193692, 6.70821487682950512251273605229, 7.01339613694661659856620335418, 7.03756303518920195876170155139