| L(s) = 1 | − 1.96·2-s + 1.85·4-s + 0.509·5-s − 3.38·7-s + 0.276·8-s − 0.999·10-s + 5.26·13-s + 6.64·14-s − 4.26·16-s + 0.741·17-s + 2.52·19-s + 0.946·20-s − 1.68·23-s − 4.74·25-s − 10.3·26-s − 6.28·28-s − 3.49·29-s − 3.49·31-s + 7.81·32-s − 1.45·34-s − 1.72·35-s + 9.04·37-s − 4.95·38-s + 0.140·40-s + 6.83·41-s + 4.35·43-s + 3.31·46-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.929·4-s + 0.227·5-s − 1.27·7-s + 0.0977·8-s − 0.316·10-s + 1.45·13-s + 1.77·14-s − 1.06·16-s + 0.179·17-s + 0.578·19-s + 0.211·20-s − 0.351·23-s − 0.948·25-s − 2.02·26-s − 1.18·28-s − 0.649·29-s − 0.628·31-s + 1.38·32-s − 0.249·34-s − 0.290·35-s + 1.48·37-s − 0.803·38-s + 0.0222·40-s + 1.06·41-s + 0.664·43-s + 0.488·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9801 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 5 | \( 1 - 0.509T + 5T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 0.741T + 17T^{2} \) |
| 19 | \( 1 - 2.52T + 19T^{2} \) |
| 23 | \( 1 + 1.68T + 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 - 6.83T + 41T^{2} \) |
| 43 | \( 1 - 4.35T + 43T^{2} \) |
| 47 | \( 1 + 9.61T + 47T^{2} \) |
| 53 | \( 1 + 0.592T + 53T^{2} \) |
| 59 | \( 1 + 4.04T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 15.4T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 2.32T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60395289027304346727074833076, −6.74127547152767957087529965782, −6.11520243889552679741661927158, −5.67725517092420638923692792578, −4.39426414039020320369684396139, −3.66721751874723247540927757600, −2.89014797345482755567688255194, −1.85746456932105724205555711016, −1.01799717087316017685168149723, 0,
1.01799717087316017685168149723, 1.85746456932105724205555711016, 2.89014797345482755567688255194, 3.66721751874723247540927757600, 4.39426414039020320369684396139, 5.67725517092420638923692792578, 6.11520243889552679741661927158, 6.74127547152767957087529965782, 7.60395289027304346727074833076
