| L(s) = 1 | − 3-s + 5-s − 0.911·7-s + 9-s − 2.91·11-s − 6.25·13-s − 15-s + 4·17-s − 19-s + 0.911·21-s − 5.34·23-s + 25-s − 27-s − 0.911·29-s + 2·31-s + 2.91·33-s − 0.911·35-s + 4.43·37-s + 6.25·39-s + 6.43·41-s − 8.25·43-s + 45-s + 5.34·47-s − 6.16·49-s − 4·51-s + 3.34·53-s − 2.91·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.344·7-s + 0.333·9-s − 0.877·11-s − 1.73·13-s − 0.258·15-s + 0.970·17-s − 0.229·19-s + 0.198·21-s − 1.11·23-s + 0.200·25-s − 0.192·27-s − 0.169·29-s + 0.359·31-s + 0.506·33-s − 0.154·35-s + 0.729·37-s + 1.00·39-s + 1.00·41-s − 1.25·43-s + 0.149·45-s + 0.779·47-s − 0.881·49-s − 0.560·51-s + 0.459·53-s − 0.392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.083966730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.083966730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 0.911T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 + 0.911T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 6.43T + 41T^{2} \) |
| 43 | \( 1 + 8.25T + 43T^{2} \) |
| 47 | \( 1 - 5.34T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 16.5T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 + 2.43T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.048988434201979413023615469983, −7.68223062242674555336878177313, −6.80629329401620300053390826437, −6.09980711429328337916898057058, −5.32183268793334977547399771920, −4.87250610194109816541097932883, −3.83358719649212264276649746141, −2.73392550128240915909749650632, −2.04416368777916527683001091977, −0.57158648188473683718577188158,
0.57158648188473683718577188158, 2.04416368777916527683001091977, 2.73392550128240915909749650632, 3.83358719649212264276649746141, 4.87250610194109816541097932883, 5.32183268793334977547399771920, 6.09980711429328337916898057058, 6.80629329401620300053390826437, 7.68223062242674555336878177313, 8.048988434201979413023615469983
