| L(s) = 1 | − 2·2-s + 2.88·3-s + 4·4-s − 5.77·6-s + 8.21·7-s − 8·8-s − 18.6·9-s − 71.5·11-s + 11.5·12-s + 66.1·13-s − 16.4·14-s + 16·16-s + 27.7·17-s + 37.3·18-s + 13.7·19-s + 23.7·21-s + 143.·22-s − 23·23-s − 23.1·24-s − 132.·26-s − 131.·27-s + 32.8·28-s + 274.·29-s + 265.·31-s − 32·32-s − 206.·33-s − 55.5·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.555·3-s + 0.5·4-s − 0.393·6-s + 0.443·7-s − 0.353·8-s − 0.690·9-s − 1.96·11-s + 0.277·12-s + 1.41·13-s − 0.313·14-s + 0.250·16-s + 0.396·17-s + 0.488·18-s + 0.166·19-s + 0.246·21-s + 1.38·22-s − 0.208·23-s − 0.196·24-s − 0.997·26-s − 0.940·27-s + 0.221·28-s + 1.75·29-s + 1.54·31-s − 0.176·32-s − 1.09·33-s − 0.280·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 2.88T + 27T^{2} \) |
| 7 | \( 1 - 8.21T + 343T^{2} \) |
| 11 | \( 1 + 71.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 27.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 274.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 265.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 204.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 69.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 187.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 135.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 502.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 161.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 103.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 985.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 817.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.11e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 400.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 485.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.03e3T + 9.12e5T^{2} \) |
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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
−8.694814688736567584109261570230, −8.226066652543623833213359930016, −7.86054134261430531530322840543, −6.58763599220616435988894194136, −5.69132484267379798196828534892, −4.77971792801442340956411962363, −3.25337416053709837774527295623, −2.64850916121615149725246579389, −1.37483762505121123588513898934, 0,
1.37483762505121123588513898934, 2.64850916121615149725246579389, 3.25337416053709837774527295623, 4.77971792801442340956411962363, 5.69132484267379798196828534892, 6.58763599220616435988894194136, 7.86054134261430531530322840543, 8.226066652543623833213359930016, 8.694814688736567584109261570230
