| L(s) = 1 | − 3.80·2-s − 6.75·3-s + 6.45·4-s − 12.6·5-s + 25.6·6-s + 5.86·8-s + 18.6·9-s + 48.2·10-s + 28.3·11-s − 43.6·12-s + 75.4·13-s + 85.6·15-s − 73.9·16-s + 119.·17-s − 71.0·18-s − 123.·19-s − 81.8·20-s − 107.·22-s + 26.7·23-s − 39.6·24-s + 35.7·25-s − 286.·26-s + 56.2·27-s − 261.·29-s − 325.·30-s − 123.·31-s + 234.·32-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 1.30·3-s + 0.807·4-s − 1.13·5-s + 1.74·6-s + 0.259·8-s + 0.691·9-s + 1.52·10-s + 0.777·11-s − 1.05·12-s + 1.60·13-s + 1.47·15-s − 1.15·16-s + 1.70·17-s − 0.930·18-s − 1.48·19-s − 0.915·20-s − 1.04·22-s + 0.242·23-s − 0.336·24-s + 0.285·25-s − 2.16·26-s + 0.400·27-s − 1.67·29-s − 1.98·30-s − 0.717·31-s + 1.29·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2107 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 + 3.80T + 8T^{2} \) |
| 3 | \( 1 + 6.75T + 27T^{2} \) |
| 5 | \( 1 + 12.6T + 125T^{2} \) |
| 11 | \( 1 - 28.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 26.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 261.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 123.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 29.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 343.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 201.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 145.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 351.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 669.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 673.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 938.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 75.0T + 7.04e5T^{2} \) |
| 97 | \( 1 - 127.T + 9.12e5T^{2} \) |
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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
−8.495210690225630206519315539151, −7.65923261995101972731002862383, −6.96592847103511742942227210204, −6.15034485369997138632579803650, −5.37658197184060576325264556156, −4.14943649942576392236828039158, −3.59524159402895205627659920327, −1.65549896841882028836594454224, −0.832650026654403881764733304232, 0,
0.832650026654403881764733304232, 1.65549896841882028836594454224, 3.59524159402895205627659920327, 4.14943649942576392236828039158, 5.37658197184060576325264556156, 6.15034485369997138632579803650, 6.96592847103511742942227210204, 7.65923261995101972731002862383, 8.495210690225630206519315539151
