L(s) = 1 | + 2·2-s − 4-s + 20·5-s − 16·7-s + 4·8-s + 40·10-s − 22·11-s + 80·13-s − 32·14-s − 19·16-s + 164·17-s − 36·19-s − 20·20-s − 44·22-s + 172·23-s + 62·25-s + 160·26-s + 16·28-s + 108·29-s − 448·31-s − 202·32-s + 328·34-s − 320·35-s + 108·37-s − 72·38-s + 80·40-s + 212·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/8·4-s + 1.78·5-s − 0.863·7-s + 0.176·8-s + 1.26·10-s − 0.603·11-s + 1.70·13-s − 0.610·14-s − 0.296·16-s + 2.33·17-s − 0.434·19-s − 0.223·20-s − 0.426·22-s + 1.55·23-s + 0.495·25-s + 1.20·26-s + 0.107·28-s + 0.691·29-s − 2.59·31-s − 1.11·32-s + 1.65·34-s − 1.54·35-s + 0.479·37-s − 0.307·38-s + 0.316·40-s + 0.807·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.821366598\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.821366598\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
good | 2 | $D_{4}$ | \( 1 - p T + 5 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 4 p T + 338 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 450 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 80 T + 4542 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 164 T + 16118 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 3242 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 172 T + 30758 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 108 T + 14062 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 448 T + 101646 T^{2} + 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 108 T + 103022 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 212 T + 148886 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 156 T + 63530 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 20 T + 202454 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 132 T - 117518 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 688 T + 404246 T^{2} - 688 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 96 T + 402398 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 448 T + 455094 T^{2} - 448 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 132 T - 187322 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 428 T + 808278 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 424 T + 1031010 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 720 T + 1262374 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1056 T + 1317010 T^{2} - 1056 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 52 T - 413466 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−13.50453805174967526545947001633, −13.26580437011742040174854867611, −12.78392508566684668116245269604, −12.59462682170779175427016595721, −11.54489047071759861418940900331, −10.79730467358998163089672901932, −10.59370404535768446421974108556, −9.838159807342146735806641667238, −9.360209932183249892951631937265, −9.058810028364249598034084213429, −8.109690982240631493201164995030, −7.43126313196315315678457759983, −6.56552416037212032264222260361, −6.03553423603517429178052398660, −5.45911315163551633951656936541, −5.16594615695106684174194864653, −3.82351732570749924444722014957, −3.32911178540281053561737129399, −2.19646146003414190013375980139, −1.13573322015703187283264619938,
1.13573322015703187283264619938, 2.19646146003414190013375980139, 3.32911178540281053561737129399, 3.82351732570749924444722014957, 5.16594615695106684174194864653, 5.45911315163551633951656936541, 6.03553423603517429178052398660, 6.56552416037212032264222260361, 7.43126313196315315678457759983, 8.109690982240631493201164995030, 9.058810028364249598034084213429, 9.360209932183249892951631937265, 9.838159807342146735806641667238, 10.59370404535768446421974108556, 10.79730467358998163089672901932, 11.54489047071759861418940900331, 12.59462682170779175427016595721, 12.78392508566684668116245269604, 13.26580437011742040174854867611, 13.50453805174967526545947001633