| L(s) = 1 | + 2-s − 0.352·3-s + 4-s + 4.32·5-s − 0.352·6-s + 7-s + 8-s − 2.87·9-s + 4.32·10-s + 2.35·11-s − 0.352·12-s + 0.801·13-s + 14-s − 1.52·15-s + 16-s + 6.22·17-s − 2.87·18-s + 4.32·20-s − 0.352·21-s + 2.35·22-s − 1.77·23-s − 0.352·24-s + 13.7·25-s + 0.801·26-s + 2.06·27-s + 28-s − 5.83·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.203·3-s + 0.5·4-s + 1.93·5-s − 0.143·6-s + 0.377·7-s + 0.353·8-s − 0.958·9-s + 1.36·10-s + 0.709·11-s − 0.101·12-s + 0.222·13-s + 0.267·14-s − 0.393·15-s + 0.250·16-s + 1.51·17-s − 0.677·18-s + 0.967·20-s − 0.0768·21-s + 0.501·22-s − 0.369·23-s − 0.0719·24-s + 2.74·25-s + 0.157·26-s + 0.398·27-s + 0.188·28-s − 1.08·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.740813060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.740813060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.352T + 3T^{2} \) |
| 5 | \( 1 - 4.32T + 5T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.801T + 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 5.83T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.64T + 41T^{2} \) |
| 43 | \( 1 - 0.873T + 43T^{2} \) |
| 47 | \( 1 + 5.83T + 47T^{2} \) |
| 53 | \( 1 + 1.12T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + 4.58T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 9.23T + 73T^{2} \) |
| 79 | \( 1 - 7.04T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + 7.29T + 89T^{2} \) |
| 97 | \( 1 + 3.05T + 97T^{2} \) |
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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
−8.314517626076476499838680070337, −7.29267829314900472609553866764, −6.44957194833157577845151288784, −5.89578856850836512420273487149, −5.47371668652524841200611848065, −4.85171032509243695526079126863, −3.63426876185883318176221287604, −2.84894352905872127055051205989, −1.94329287585814635329165025848, −1.20070304511893480588138578214,
1.20070304511893480588138578214, 1.94329287585814635329165025848, 2.84894352905872127055051205989, 3.63426876185883318176221287604, 4.85171032509243695526079126863, 5.47371668652524841200611848065, 5.89578856850836512420273487149, 6.44957194833157577845151288784, 7.29267829314900472609553866764, 8.314517626076476499838680070337
