| L(s) = 1 | − 3.21·2-s + 2.32·4-s + 13.8·5-s + 29.9·7-s + 18.2·8-s − 44.5·10-s + 38.5·11-s − 23.0·13-s − 96.2·14-s − 77.1·16-s − 22.8·17-s + 2.58·19-s + 32.2·20-s − 123.·22-s − 49.7·23-s + 67.3·25-s + 74.0·26-s + 69.6·28-s + 235.·29-s − 309.·31-s + 102.·32-s + 73.5·34-s + 415.·35-s − 239.·37-s − 8.29·38-s + 252.·40-s − 228.·41-s + ⋯ |
| L(s) = 1 | − 1.13·2-s + 0.290·4-s + 1.24·5-s + 1.61·7-s + 0.806·8-s − 1.40·10-s + 1.05·11-s − 0.491·13-s − 1.83·14-s − 1.20·16-s − 0.326·17-s + 0.0311·19-s + 0.360·20-s − 1.20·22-s − 0.451·23-s + 0.538·25-s + 0.558·26-s + 0.469·28-s + 1.51·29-s − 1.79·31-s + 0.563·32-s + 0.371·34-s + 2.00·35-s − 1.06·37-s − 0.0354·38-s + 0.999·40-s − 0.872·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.992383311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992383311\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 3.21T + 8T^{2} \) |
| 5 | \( 1 - 13.8T + 125T^{2} \) |
| 7 | \( 1 - 29.9T + 343T^{2} \) |
| 11 | \( 1 - 38.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.58T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 309.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 239.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 228.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 267.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 837.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 894.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 881.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 271.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 706.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.49e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.76e3T + 9.12e5T^{2} \) |
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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
−8.857735664739793637092381918913, −8.203714287363369065851448404578, −7.31518476812576371818456060302, −6.62884779299328374685529278812, −5.47747009965384569323105020771, −4.87570724182492419539996804803, −3.92609928418142131500244781411, −2.13519486348840234614666271392, −1.75239888663152523611886966883, −0.811013686529225260477292088276,
0.811013686529225260477292088276, 1.75239888663152523611886966883, 2.13519486348840234614666271392, 3.92609928418142131500244781411, 4.87570724182492419539996804803, 5.47747009965384569323105020771, 6.62884779299328374685529278812, 7.31518476812576371818456060302, 8.203714287363369065851448404578, 8.857735664739793637092381918913
