| L(s) = 1 | − 2-s + 1.87·3-s + 4-s − 1.87·6-s − 8-s + 2.53·9-s + 0.347·11-s + 1.87·12-s + 16-s − 17-s − 2.53·18-s − 0.347·22-s − 1.87·24-s + 25-s + 2.87·27-s − 32-s + 0.652·33-s + 34-s + 2.53·36-s − 1.53·41-s − 43-s + 0.347·44-s + 1.87·48-s + 49-s − 50-s − 1.87·51-s − 2.87·54-s + ⋯ |
| L(s) = 1 | − 2-s + 1.87·3-s + 4-s − 1.87·6-s − 8-s + 2.53·9-s + 0.347·11-s + 1.87·12-s + 16-s − 17-s − 2.53·18-s − 0.347·22-s − 1.87·24-s + 25-s + 2.87·27-s − 32-s + 0.652·33-s + 34-s + 2.53·36-s − 1.53·41-s − 43-s + 0.347·44-s + 1.87·48-s + 49-s − 50-s − 1.87·51-s − 2.87·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.614998137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614998137\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.87T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.347T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.53T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 0.347T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - 1.53T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + 1.53T + T^{2} \) |
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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
−8.886969300371726701307961693045, −8.428586371607409377478610538836, −7.69045723559412709775374601597, −6.98119235312974359389896365805, −6.42733388039291255231291936976, −4.91259961683046460173713252046, −3.84078533487694991834191055360, −3.06077957811915668233296723724, −2.27439455309821519325338217178, −1.42546791126607998135656767353,
1.42546791126607998135656767353, 2.27439455309821519325338217178, 3.06077957811915668233296723724, 3.84078533487694991834191055360, 4.91259961683046460173713252046, 6.42733388039291255231291936976, 6.98119235312974359389896365805, 7.69045723559412709775374601597, 8.428586371607409377478610538836, 8.886969300371726701307961693045
