| L(s) = 1 | + 162·9-s − 1.31e3·11-s − 2.72e3·19-s + 4.43e3·29-s + 3.40e3·31-s − 3.63e3·41-s − 2.49e4·49-s + 1.73e4·59-s − 6.93e4·61-s − 1.89e3·71-s + 9.30e4·79-s − 3.28e4·81-s + 2.09e5·89-s − 2.12e5·99-s + 8.49e4·101-s + 2.90e5·109-s + 9.68e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.00e5·169-s + ⋯ |
| L(s) = 1 | + 2/3·9-s − 3.26·11-s − 1.73·19-s + 0.979·29-s + 0.635·31-s − 0.337·41-s − 1.48·49-s + 0.648·59-s − 2.38·61-s − 0.0446·71-s + 1.67·79-s − 5/9·81-s + 2.80·89-s − 2.17·99-s + 0.828·101-s + 2.33·109-s + 6.01·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s + 1.88·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.05134201104\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05134201104\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 p^{4} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 24950 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 656 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 700150 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16386 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 1364 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 8041482 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2218 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 1700 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 137972198 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 1818 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 183051730 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 312908538 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 225456410 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8668 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 34670 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 437725858 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 948 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 164280782 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 46536 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 3451998 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 104934 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 15860327998 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.48412043892979570426331620970, −10.31172682188460086486590354602, −10.09996619736733656994003085014, −9.359928473156736680533288580607, −8.712647417952766837336505650084, −8.334856608351929095004169391129, −7.74230748727845774518254614886, −7.73979829652982475043249870085, −7.01720508655105318316414923753, −6.20490876722207459597183547641, −6.16090953726794671604024093392, −5.19302384307400839565448288388, −4.71916898369486766200969715915, −4.71120428365409301494224423758, −3.63808181675809306608218983480, −3.02133857873629167232537570509, −2.37443700297175720114766246713, −2.09997337865011523416872817435, −1.05551199823078966696348272244, −0.05943118198874736705756973704,
0.05943118198874736705756973704, 1.05551199823078966696348272244, 2.09997337865011523416872817435, 2.37443700297175720114766246713, 3.02133857873629167232537570509, 3.63808181675809306608218983480, 4.71120428365409301494224423758, 4.71916898369486766200969715915, 5.19302384307400839565448288388, 6.16090953726794671604024093392, 6.20490876722207459597183547641, 7.01720508655105318316414923753, 7.73979829652982475043249870085, 7.74230748727845774518254614886, 8.334856608351929095004169391129, 8.712647417952766837336505650084, 9.359928473156736680533288580607, 10.09996619736733656994003085014, 10.31172682188460086486590354602, 10.48412043892979570426331620970
