L(s) = 1 | + 0.414·3-s + 5-s − 2.82·9-s + 0.828·11-s − 4.82·13-s + 0.414·15-s + 4.82·17-s − 2.82·19-s − 0.414·23-s + 25-s − 2.41·27-s − 29-s + 6·31-s + 0.343·33-s − 1.99·39-s − 7.82·41-s − 3.58·43-s − 2.82·45-s − 2·47-s + 1.99·51-s − 1.17·53-s + 0.828·55-s − 1.17·57-s − 4.48·59-s + 5.48·61-s − 4.82·65-s − 9.58·67-s + ⋯ |
L(s) = 1 | + 0.239·3-s + 0.447·5-s − 0.942·9-s + 0.249·11-s − 1.33·13-s + 0.106·15-s + 1.17·17-s − 0.648·19-s − 0.0863·23-s + 0.200·25-s − 0.464·27-s − 0.185·29-s + 1.07·31-s + 0.0597·33-s − 0.320·39-s − 1.22·41-s − 0.546·43-s − 0.421·45-s − 0.291·47-s + 0.280·51-s − 0.160·53-s + 0.111·55-s − 0.155·57-s − 0.583·59-s + 0.702·61-s − 0.598·65-s − 1.17·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 0.414T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 - 5.48T + 61T^{2} \) |
| 67 | \( 1 + 9.58T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 0.828T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 8.65T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.206011324173688267594849797163, −7.40708693989609610217642267455, −6.62641534382088283647335788525, −5.81033451226306419630331086382, −5.18661549011192502982190408329, −4.33786214732182446865009859525, −3.19633947561842943398030785236, −2.59942774280718036414860003182, −1.54042571092675288000290094150, 0,
1.54042571092675288000290094150, 2.59942774280718036414860003182, 3.19633947561842943398030785236, 4.33786214732182446865009859525, 5.18661549011192502982190408329, 5.81033451226306419630331086382, 6.62641534382088283647335788525, 7.40708693989609610217642267455, 8.206011324173688267594849797163