| L(s) = 1 | − 3.76·2-s + 3.72·3-s + 6.20·4-s − 5·5-s − 14.0·6-s − 30.0·7-s + 6.77·8-s − 13.1·9-s + 18.8·10-s + 7.58·11-s + 23.0·12-s − 13·13-s + 113.·14-s − 18.6·15-s − 75.1·16-s + 49.1·17-s + 49.4·18-s + 31.8·19-s − 31.0·20-s − 111.·21-s − 28.6·22-s + 96.1·23-s + 25.2·24-s + 25·25-s + 48.9·26-s − 149.·27-s − 186.·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 0.716·3-s + 0.775·4-s − 0.447·5-s − 0.954·6-s − 1.62·7-s + 0.299·8-s − 0.486·9-s + 0.595·10-s + 0.208·11-s + 0.555·12-s − 0.277·13-s + 2.15·14-s − 0.320·15-s − 1.17·16-s + 0.701·17-s + 0.647·18-s + 0.384·19-s − 0.346·20-s − 1.16·21-s − 0.277·22-s + 0.871·23-s + 0.214·24-s + 0.200·25-s + 0.369·26-s − 1.06·27-s − 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 + 3.76T + 8T^{2} \) |
| 3 | \( 1 - 3.72T + 27T^{2} \) |
| 7 | \( 1 + 30.0T + 343T^{2} \) |
| 11 | \( 1 - 7.58T + 1.33e3T^{2} \) |
| 17 | \( 1 - 49.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 31.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 136.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 338.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 379.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 198.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 26.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 909.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 387.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.61e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 344.T + 9.12e5T^{2} \) |
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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
−8.562459810964203923894952846114, −7.75211219603151430988072447619, −7.20637194666580391960434835609, −6.36644312051399420410847864418, −5.31855467955138832364065397863, −3.91918827522377907874425765093, −3.19619252903320393524648903667, −2.34380338920193773339146844724, −0.908364734598880880406422441000, 0,
0.908364734598880880406422441000, 2.34380338920193773339146844724, 3.19619252903320393524648903667, 3.91918827522377907874425765093, 5.31855467955138832364065397863, 6.36644312051399420410847864418, 7.20637194666580391960434835609, 7.75211219603151430988072447619, 8.562459810964203923894952846114
