| L(s) = 1 | − 3·3-s − 4-s + 3·8-s + 2·9-s − 8·11-s + 3·12-s + 5·13-s − 16-s + 7·17-s + 13·19-s − 23-s − 9·24-s − 15·25-s + 4·27-s − 2·31-s − 6·32-s + 24·33-s − 2·36-s − 13·37-s − 15·39-s − 3·41-s − 43-s + 8·44-s − 9·47-s + 3·48-s − 21·51-s − 5·52-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1/2·4-s + 1.06·8-s + 2/3·9-s − 2.41·11-s + 0.866·12-s + 1.38·13-s − 1/4·16-s + 1.69·17-s + 2.98·19-s − 0.208·23-s − 1.83·24-s − 3·25-s + 0.769·27-s − 0.359·31-s − 1.06·32-s + 4.17·33-s − 1/3·36-s − 2.13·37-s − 2.40·39-s − 0.468·41-s − 0.152·43-s + 1.20·44-s − 1.31·47-s + 0.433·48-s − 2.94·51-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7303556936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7303556936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} - 3 T^{3} + p T^{4} + p^{3} T^{6} \) |
| 3 | $S_4\times C_2$ | \( 1 + p T + 7 T^{2} + 11 T^{3} + 7 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 37 T^{2} + 136 T^{3} + 37 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 21 T^{2} - 67 T^{3} + 21 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 59 T^{2} - 231 T^{3} + 59 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 13 T + 103 T^{2} - 543 T^{3} + 103 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + T + 49 T^{2} + 71 T^{3} + 49 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 67 T^{2} - 24 T^{3} + 67 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 25 T^{2} + 180 T^{3} + 25 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 13 T + 141 T^{2} + 953 T^{3} + 141 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + T + 121 T^{2} + 83 T^{3} + 121 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 9 T + 129 T^{2} + 713 T^{3} + 129 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 8 T + 163 T^{2} + 808 T^{3} + 163 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 85 T^{2} + 640 T^{3} + 85 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T - 17 T^{2} - 444 T^{3} - 17 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 125 T^{2} - 8 p T^{3} + 125 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 205 T^{2} + 1220 T^{3} + 205 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 147 T^{2} + 956 T^{3} + 147 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 281 T^{2} - 2268 T^{3} + 281 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 5 T + 249 T^{2} - 827 T^{3} + 249 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 19 T + 387 T^{2} - 3749 T^{3} + 387 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.108778248243266221543607618883, −7.67536256495095490403583237224, −7.61888424243287953806917782149, −7.56844208204256961303118428006, −7.13082643109043463860779240304, −6.78902796673198617108705211862, −6.46275403261896983250122787352, −5.95436469359043940552632019504, −5.76476478707894776188718291553, −5.74101432670967565354028464396, −5.41582365237570744644510117108, −5.16007147633462664088036963990, −5.11025189147687812332511829522, −4.82350233916463722790187803215, −4.22248495341417712037576700795, −4.07180317785127454839367233571, −3.58195727433523434818337767375, −3.23655563653654186676139364623, −3.04375578964435953909760925159, −2.88307662393173824627902503160, −1.82744876008328932777988601048, −1.64348191700167938959836211436, −1.58376281372782804575721838264, −0.58950026196098288091025239744, −0.37681834620498450799425196019,
0.37681834620498450799425196019, 0.58950026196098288091025239744, 1.58376281372782804575721838264, 1.64348191700167938959836211436, 1.82744876008328932777988601048, 2.88307662393173824627902503160, 3.04375578964435953909760925159, 3.23655563653654186676139364623, 3.58195727433523434818337767375, 4.07180317785127454839367233571, 4.22248495341417712037576700795, 4.82350233916463722790187803215, 5.11025189147687812332511829522, 5.16007147633462664088036963990, 5.41582365237570744644510117108, 5.74101432670967565354028464396, 5.76476478707894776188718291553, 5.95436469359043940552632019504, 6.46275403261896983250122787352, 6.78902796673198617108705211862, 7.13082643109043463860779240304, 7.56844208204256961303118428006, 7.61888424243287953806917782149, 7.67536256495095490403583237224, 8.108778248243266221543607618883
