| L(s) = 1 | − 2-s − 3-s + 4-s − 4.07·5-s + 6-s + 2.58·7-s − 8-s + 9-s + 4.07·10-s + 3.75·11-s − 12-s − 13-s − 2.58·14-s + 4.07·15-s + 16-s − 4.96·17-s − 18-s + 3.27·19-s − 4.07·20-s − 2.58·21-s − 3.75·22-s + 6.21·23-s + 24-s + 11.5·25-s + 26-s − 27-s + 2.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.82·5-s + 0.408·6-s + 0.975·7-s − 0.353·8-s + 0.333·9-s + 1.28·10-s + 1.13·11-s − 0.288·12-s − 0.277·13-s − 0.689·14-s + 1.05·15-s + 0.250·16-s − 1.20·17-s − 0.235·18-s + 0.751·19-s − 0.910·20-s − 0.563·21-s − 0.799·22-s + 1.29·23-s + 0.204·24-s + 2.31·25-s + 0.196·26-s − 0.192·27-s + 0.487·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8952642113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8952642113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 4.07T + 5T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 - 3.75T + 11T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 + 3.00T + 29T^{2} \) |
| 31 | \( 1 - 0.515T + 31T^{2} \) |
| 37 | \( 1 + 0.169T + 37T^{2} \) |
| 41 | \( 1 - 9.14T + 41T^{2} \) |
| 43 | \( 1 - 5.16T + 43T^{2} \) |
| 47 | \( 1 - 9.19T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 + 7.18T + 59T^{2} \) |
| 61 | \( 1 - 8.72T + 61T^{2} \) |
| 67 | \( 1 - 5.57T + 67T^{2} \) |
| 71 | \( 1 + 8.81T + 71T^{2} \) |
| 73 | \( 1 + 6.75T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 1.76T + 97T^{2} \) |
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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
−7.68981342045412783052093976653, −7.28023012081404445850524595171, −6.81234230478030247667998378637, −5.83221170652372456412229764036, −4.83188675844358971533724373805, −4.33895233398318456250178668233, −3.65985254629091781670639403785, −2.60908471890799558177434732886, −1.34858878400570633391812850815, −0.60408568698448996241862773379,
0.60408568698448996241862773379, 1.34858878400570633391812850815, 2.60908471890799558177434732886, 3.65985254629091781670639403785, 4.33895233398318456250178668233, 4.83188675844358971533724373805, 5.83221170652372456412229764036, 6.81234230478030247667998378637, 7.28023012081404445850524595171, 7.68981342045412783052093976653
