| L(s) = 1 | − 3.41·5-s − 1.12·7-s + 7.01·13-s − 4.49·17-s − 5.53·19-s − 6.03·23-s + 6.64·25-s + 1.51·29-s − 3.29·31-s + 3.84·35-s + 7.26·37-s + 9.48·41-s − 1.96·43-s − 6.08·47-s − 5.73·49-s − 1.17·53-s − 3.46·59-s − 2.99·61-s − 23.9·65-s + 11.1·67-s − 16.2·71-s − 4.18·73-s − 2.31·79-s + 0.567·83-s + 15.3·85-s + 13.9·89-s − 7.90·91-s + ⋯ |
| L(s) = 1 | − 1.52·5-s − 0.425·7-s + 1.94·13-s − 1.08·17-s − 1.27·19-s − 1.25·23-s + 1.32·25-s + 0.282·29-s − 0.591·31-s + 0.649·35-s + 1.19·37-s + 1.48·41-s − 0.300·43-s − 0.887·47-s − 0.818·49-s − 0.161·53-s − 0.450·59-s − 0.383·61-s − 2.96·65-s + 1.36·67-s − 1.92·71-s − 0.490·73-s − 0.260·79-s + 0.0623·83-s + 1.66·85-s + 1.47·89-s − 0.828·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8333434324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8333434324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 13 | \( 1 - 7.01T + 13T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 6.03T + 23T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 + 3.29T + 31T^{2} \) |
| 37 | \( 1 - 7.26T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 + 1.96T + 43T^{2} \) |
| 47 | \( 1 + 6.08T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 16.2T + 71T^{2} \) |
| 73 | \( 1 + 4.18T + 73T^{2} \) |
| 79 | \( 1 + 2.31T + 79T^{2} \) |
| 83 | \( 1 - 0.567T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.927128657657073218621912238762, −7.10498890048033073412508672296, −6.23377419866158042009976417306, −6.08670660902817685671288204769, −4.67696441114498172182119820745, −4.08573728824749541683143013748, −3.71976382183839360096114280289, −2.80723909332936898263240542795, −1.68224335061794169213099577453, −0.44111557181197106447092434271,
0.44111557181197106447092434271, 1.68224335061794169213099577453, 2.80723909332936898263240542795, 3.71976382183839360096114280289, 4.08573728824749541683143013748, 4.67696441114498172182119820745, 6.08670660902817685671288204769, 6.23377419866158042009976417306, 7.10498890048033073412508672296, 7.927128657657073218621912238762
