L(s) = 1 | + 2-s − 2.42·3-s + 4-s − 5-s − 2.42·6-s − 7-s + 8-s + 2.85·9-s − 10-s − 0.0639·11-s − 2.42·12-s − 0.278·13-s − 14-s + 2.42·15-s + 16-s + 3.33·17-s + 2.85·18-s + 3.28·19-s − 20-s + 2.42·21-s − 0.0639·22-s − 5.13·23-s − 2.42·24-s + 25-s − 0.278·26-s + 0.347·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.447·5-s − 0.987·6-s − 0.377·7-s + 0.353·8-s + 0.952·9-s − 0.316·10-s − 0.0192·11-s − 0.698·12-s − 0.0773·13-s − 0.267·14-s + 0.624·15-s + 0.250·16-s + 0.808·17-s + 0.673·18-s + 0.754·19-s − 0.223·20-s + 0.528·21-s − 0.0136·22-s − 1.07·23-s − 0.493·24-s + 0.200·25-s − 0.0546·26-s + 0.0668·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 71 | \( 1 - T \) |
good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 11 | \( 1 + 0.0639T + 11T^{2} \) |
| 13 | \( 1 + 0.278T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 - 3.28T + 19T^{2} \) |
| 23 | \( 1 + 5.13T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 + 1.13T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 - 0.166T + 41T^{2} \) |
| 43 | \( 1 - 3.09T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 73 | \( 1 + 0.812T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 - 0.355T + 83T^{2} \) |
| 89 | \( 1 + 2.75T + 89T^{2} \) |
| 97 | \( 1 + 7.43T + 97T^{2} \) |
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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
−7.44843624704526123897112062316, −7.14337358632008679445031253726, −6.20563308195759526338516710542, −5.50890517553409476555013384066, −5.30523350178785803101230112668, −4.10924026216584474957632040672, −3.66224235314226211813074331739, −2.49854272344665814847869680564, −1.19964713376293945521917128654, 0,
1.19964713376293945521917128654, 2.49854272344665814847869680564, 3.66224235314226211813074331739, 4.10924026216584474957632040672, 5.30523350178785803101230112668, 5.50890517553409476555013384066, 6.20563308195759526338516710542, 7.14337358632008679445031253726, 7.44843624704526123897112062316