| L(s) = 1 | − 3·2-s + 4-s + 5·5-s + 21·8-s − 15·10-s + 45·11-s − 31·13-s − 71·16-s − 96·17-s + 149·19-s + 5·20-s − 135·22-s + 141·23-s + 25·25-s + 93·26-s − 48·29-s − 178·31-s + 45·32-s + 288·34-s + 371·37-s − 447·38-s + 105·40-s − 225·41-s + 344·43-s + 45·44-s − 423·46-s − 375·47-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 1/8·4-s + 0.447·5-s + 0.928·8-s − 0.474·10-s + 1.23·11-s − 0.661·13-s − 1.10·16-s − 1.36·17-s + 1.79·19-s + 0.0559·20-s − 1.30·22-s + 1.27·23-s + 1/5·25-s + 0.701·26-s − 0.307·29-s − 1.03·31-s + 0.248·32-s + 1.45·34-s + 1.64·37-s − 1.90·38-s + 0.415·40-s − 0.857·41-s + 1.21·43-s + 0.154·44-s − 1.35·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.345960201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345960201\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 T + p^{3} T^{2} \) |
| 11 | \( 1 - 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 31 T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 - 149 T + p^{3} T^{2} \) |
| 23 | \( 1 - 141 T + p^{3} T^{2} \) |
| 29 | \( 1 + 48 T + p^{3} T^{2} \) |
| 31 | \( 1 + 178 T + p^{3} T^{2} \) |
| 37 | \( 1 - 371 T + p^{3} T^{2} \) |
| 41 | \( 1 + 225 T + p^{3} T^{2} \) |
| 43 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 375 T + p^{3} T^{2} \) |
| 53 | \( 1 - 663 T + p^{3} T^{2} \) |
| 59 | \( 1 - 60 T + p^{3} T^{2} \) |
| 61 | \( 1 - 392 T + p^{3} T^{2} \) |
| 67 | \( 1 + 280 T + p^{3} T^{2} \) |
| 71 | \( 1 + 258 T + p^{3} T^{2} \) |
| 73 | \( 1 - 578 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 - 432 T + p^{3} T^{2} \) |
| 89 | \( 1 - 234 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1352 T + p^{3} T^{2} \) |
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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
−9.091460058918671953693876445177, −8.040976219492954756874590803984, −7.16373994102285648683195644896, −6.74963619206701239733832117425, −5.53003508629502893187627280872, −4.73447526785219955710349268869, −3.81516346547706635399558955097, −2.54949408078808590207912214958, −1.49084657608618105321264622275, −0.67067631501041024230593589847,
0.67067631501041024230593589847, 1.49084657608618105321264622275, 2.54949408078808590207912214958, 3.81516346547706635399558955097, 4.73447526785219955710349268869, 5.53003508629502893187627280872, 6.74963619206701239733832117425, 7.16373994102285648683195644896, 8.040976219492954756874590803984, 9.091460058918671953693876445177
