| L(s) = 1 | − 2-s + 4-s − 1.39·5-s − 3.18·7-s − 8-s + 1.39·10-s − 0.323·11-s + 2.32·13-s + 3.18·14-s + 16-s − 1.72·17-s − 2.86·19-s − 1.39·20-s + 0.323·22-s + 6.50·23-s − 3.04·25-s − 2.32·26-s − 3.18·28-s − 0.0643·29-s + 5.10·31-s − 32-s + 1.72·34-s + 4.45·35-s + 8.98·37-s + 2.86·38-s + 1.39·40-s + 2.65·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.625·5-s − 1.20·7-s − 0.353·8-s + 0.442·10-s − 0.0975·11-s + 0.644·13-s + 0.851·14-s + 0.250·16-s − 0.417·17-s − 0.656·19-s − 0.312·20-s + 0.0689·22-s + 1.35·23-s − 0.609·25-s − 0.455·26-s − 0.601·28-s − 0.0119·29-s + 0.917·31-s − 0.176·32-s + 0.295·34-s + 0.752·35-s + 1.47·37-s + 0.464·38-s + 0.221·40-s + 0.414·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7803547764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7803547764\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 + 0.323T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 1.72T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 + 0.0643T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 - 2.65T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 4.79T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 67 | \( 1 + 4.69T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−9.651277935830380482623882154380, −9.150524933110662642054948910507, −8.248342970226689120037345636894, −7.46969977897348269598395783305, −6.55103099382117253862630666796, −5.96953127185242466684810529717, −4.48430178871476826379961281754, −3.47837216586679871804272213063, −2.50191690690808572462807579836, −0.73014257456017341271987376300,
0.73014257456017341271987376300, 2.50191690690808572462807579836, 3.47837216586679871804272213063, 4.48430178871476826379961281754, 5.96953127185242466684810529717, 6.55103099382117253862630666796, 7.46969977897348269598395783305, 8.248342970226689120037345636894, 9.150524933110662642054948910507, 9.651277935830380482623882154380
