| L(s) = 1 | + 2·2-s − 5.04·3-s + 4·4-s + 5·5-s − 10.0·6-s + 5.03·7-s + 8·8-s − 1.58·9-s + 10·10-s − 5.58·11-s − 20.1·12-s + 62.7·13-s + 10.0·14-s − 25.2·15-s + 16·16-s − 19.7·17-s − 3.17·18-s + 158.·19-s + 20·20-s − 25.3·21-s − 11.1·22-s − 23·23-s − 40.3·24-s + 25·25-s + 125.·26-s + 144.·27-s + 20.1·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.970·3-s + 0.5·4-s + 0.447·5-s − 0.685·6-s + 0.271·7-s + 0.353·8-s − 0.0588·9-s + 0.316·10-s − 0.153·11-s − 0.485·12-s + 1.33·13-s + 0.192·14-s − 0.433·15-s + 0.250·16-s − 0.281·17-s − 0.0416·18-s + 1.91·19-s + 0.223·20-s − 0.263·21-s − 0.108·22-s − 0.208·23-s − 0.342·24-s + 0.200·25-s + 0.946·26-s + 1.02·27-s + 0.135·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.290227740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.290227740\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 5.04T + 27T^{2} \) |
| 7 | \( 1 - 5.03T + 343T^{2} \) |
| 11 | \( 1 + 5.58T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 19.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 35.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 282.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 139.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 436.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 330.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 568.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 85.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 369.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 310.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 158.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 106.T + 9.12e5T^{2} \) |
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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
−11.67345530762632733283283772555, −11.11020120116598408422340923262, −10.15679807984091339771532793050, −8.829961703329520723342885000628, −7.50403396116084908236203261108, −6.20044438450013697126328055703, −5.65739164627286368975954266349, −4.55781413775353385474541713235, −3.02282063926065020601473323021, −1.16057506792724057477636876681,
1.16057506792724057477636876681, 3.02282063926065020601473323021, 4.55781413775353385474541713235, 5.65739164627286368975954266349, 6.20044438450013697126328055703, 7.50403396116084908236203261108, 8.829961703329520723342885000628, 10.15679807984091339771532793050, 11.11020120116598408422340923262, 11.67345530762632733283283772555
