| L(s) = 1 | + 1.24·2-s + 0.554·4-s + 5-s − 0.445·7-s − 0.554·8-s + 9-s + 1.24·10-s − 1.80·11-s − 0.554·14-s − 1.24·16-s − 1.80·17-s + 1.24·18-s + 1.24·19-s + 0.554·20-s − 2.24·22-s + 25-s − 0.246·28-s + 1.24·29-s − 0.999·32-s − 2.24·34-s − 0.445·35-s + 0.554·36-s + 1.55·38-s − 0.554·40-s − 0.445·41-s − 1.80·43-s − 0.999·44-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.554·4-s + 5-s − 0.445·7-s − 0.554·8-s + 9-s + 1.24·10-s − 1.80·11-s − 0.554·14-s − 1.24·16-s − 1.80·17-s + 1.24·18-s + 1.24·19-s + 0.554·20-s − 2.24·22-s + 25-s − 0.246·28-s + 1.24·29-s − 0.999·32-s − 2.24·34-s − 0.445·35-s + 0.554·36-s + 1.55·38-s − 0.554·40-s − 0.445·41-s − 1.80·43-s − 0.999·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.594961791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.594961791\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 107 | \( 1 - T \) |
| good | 2 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 7 | \( 1 + 0.445T + T^{2} \) |
| 11 | \( 1 + 1.80T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.80T + T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - 1.24T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - 1.24T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.24T + T^{2} \) |
| 79 | \( 1 + 0.445T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.24T + T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−11.09542043425084151235893606274, −10.16491969189855020367638066021, −9.527908122818056024124474424087, −8.388477665810043337945709438901, −7.00750874914935526178488014842, −6.31872134854547838858742969326, −5.17162941320833479001921604540, −4.71807635874933895606621314285, −3.23337952857169682004924550635, −2.25262003142672894754897117258,
2.25262003142672894754897117258, 3.23337952857169682004924550635, 4.71807635874933895606621314285, 5.17162941320833479001921604540, 6.31872134854547838858742969326, 7.00750874914935526178488014842, 8.388477665810043337945709438901, 9.527908122818056024124474424087, 10.16491969189855020367638066021, 11.09542043425084151235893606274
