| L(s) = 1 | + 4.08·2-s + 3·3-s + 8.68·4-s − 7.45·5-s + 12.2·6-s + 34.0·7-s + 2.80·8-s + 9·9-s − 30.4·10-s + 30.1·11-s + 26.0·12-s − 5.30·13-s + 138.·14-s − 22.3·15-s − 58.0·16-s + 63.2·17-s + 36.7·18-s − 86.0·19-s − 64.7·20-s + 102.·21-s + 122.·22-s + 83.0·23-s + 8.40·24-s − 69.4·25-s − 21.6·26-s + 27·27-s + 295.·28-s + ⋯ |
| L(s) = 1 | + 1.44·2-s + 0.577·3-s + 1.08·4-s − 0.666·5-s + 0.833·6-s + 1.83·7-s + 0.123·8-s + 0.333·9-s − 0.963·10-s + 0.825·11-s + 0.626·12-s − 0.113·13-s + 2.65·14-s − 0.384·15-s − 0.906·16-s + 0.902·17-s + 0.481·18-s − 1.03·19-s − 0.723·20-s + 1.06·21-s + 1.19·22-s + 0.752·23-s + 0.0714·24-s − 0.555·25-s − 0.163·26-s + 0.192·27-s + 1.99·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.461356772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.461356772\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
| good | 2 | \( 1 - 4.08T + 8T^{2} \) |
| 5 | \( 1 + 7.45T + 125T^{2} \) |
| 7 | \( 1 - 34.0T + 343T^{2} \) |
| 11 | \( 1 - 30.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 5.30T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 83.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 358.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 139.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 608.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 361.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 519.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 733.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 280.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 239.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 149.T + 9.12e5T^{2} \) |
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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
−12.13572491008668159417956013683, −11.72401769192977958906747378300, −10.66433590378710891815346462752, −8.917495881411767829502829207921, −8.035164094152241681789540016139, −6.92190928331969405689673671612, −5.35860817900245858538110485146, −4.43475181838014569196791164359, −3.52697469299822199448925613870, −1.82446042836825657889543150674,
1.82446042836825657889543150674, 3.52697469299822199448925613870, 4.43475181838014569196791164359, 5.35860817900245858538110485146, 6.92190928331969405689673671612, 8.035164094152241681789540016139, 8.917495881411767829502829207921, 10.66433590378710891815346462752, 11.72401769192977958906747378300, 12.13572491008668159417956013683
