| L(s) = 1 | + 2-s + 4-s − 5.12·7-s + 8-s − 2·11-s − 4·13-s − 5.12·14-s + 16-s − 5.12·17-s − 19-s − 2·22-s − 4·26-s − 5.12·28-s + 2·29-s + 0.876·31-s + 32-s − 5.12·34-s − 38-s − 3.12·41-s + 6.24·43-s − 2·44-s − 6.24·47-s + 19.2·49-s − 4·52-s + 12.2·53-s − 5.12·56-s + 2·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.93·7-s + 0.353·8-s − 0.603·11-s − 1.10·13-s − 1.36·14-s + 0.250·16-s − 1.24·17-s − 0.229·19-s − 0.426·22-s − 0.784·26-s − 0.968·28-s + 0.371·29-s + 0.157·31-s + 0.176·32-s − 0.878·34-s − 0.162·38-s − 0.487·41-s + 0.952·43-s − 0.301·44-s − 0.911·47-s + 2.74·49-s − 0.554·52-s + 1.68·53-s − 0.684·56-s + 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.349784497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349784497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 0.876T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 5.12T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 0.876T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.50485618074827769987458506476, −6.87391342217408586087165476868, −6.46618467384268748734757919467, −5.74737208757160195174619815062, −4.97462871060746259096820064116, −4.23347232357566891985881686359, −3.45849830197007610315200630290, −2.70222716331543466444033038520, −2.21562577161897200636248566516, −0.46751258834196722236257667205,
0.46751258834196722236257667205, 2.21562577161897200636248566516, 2.70222716331543466444033038520, 3.45849830197007610315200630290, 4.23347232357566891985881686359, 4.97462871060746259096820064116, 5.74737208757160195174619815062, 6.46618467384268748734757919467, 6.87391342217408586087165476868, 7.50485618074827769987458506476
