| L(s) = 1 | + 1.45·2-s − 8.28·3-s − 5.87·4-s + 5·5-s − 12.0·6-s + 3.32·7-s − 20.2·8-s + 41.6·9-s + 7.28·10-s + 11·11-s + 48.6·12-s − 21.6·13-s + 4.84·14-s − 41.4·15-s + 17.5·16-s − 1.48·17-s + 60.6·18-s − 19·19-s − 29.3·20-s − 27.5·21-s + 16.0·22-s − 141.·23-s + 167.·24-s + 25·25-s − 31.5·26-s − 121.·27-s − 19.5·28-s + ⋯ |
| L(s) = 1 | + 0.515·2-s − 1.59·3-s − 0.734·4-s + 0.447·5-s − 0.821·6-s + 0.179·7-s − 0.893·8-s + 1.54·9-s + 0.230·10-s + 0.301·11-s + 1.17·12-s − 0.461·13-s + 0.0924·14-s − 0.713·15-s + 0.273·16-s − 0.0212·17-s + 0.794·18-s − 0.229·19-s − 0.328·20-s − 0.285·21-s + 0.155·22-s − 1.28·23-s + 1.42·24-s + 0.200·25-s − 0.238·26-s − 0.863·27-s − 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7207415575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7207415575\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 1.45T + 8T^{2} \) |
| 3 | \( 1 + 8.28T + 27T^{2} \) |
| 7 | \( 1 - 3.32T + 343T^{2} \) |
| 13 | \( 1 + 21.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 1.48T + 4.91e3T^{2} \) |
| 23 | \( 1 + 141.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 252.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 14.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 150.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 219.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 309.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 237.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 678.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 394.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 205.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 414.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 481.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 887.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 328.T + 9.12e5T^{2} \) |
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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
−9.750486357264138033020460042280, −8.884800460907806800809617403409, −7.72176963597880362753149660450, −6.61505739076065323047982272600, −5.89375968136500367788775117740, −5.28575775644718179486353573653, −4.55366977494562640574291287813, −3.61755152740108166362996606957, −1.86626850125693486534638868690, −0.44771907549053438277307270551,
0.44771907549053438277307270551, 1.86626850125693486534638868690, 3.61755152740108166362996606957, 4.55366977494562640574291287813, 5.28575775644718179486353573653, 5.89375968136500367788775117740, 6.61505739076065323047982272600, 7.72176963597880362753149660450, 8.884800460907806800809617403409, 9.750486357264138033020460042280
