| L(s) = 1 | + 0.850·2-s − 7.27·4-s − 12.0·5-s − 14.5·7-s − 12.9·8-s − 10.2·10-s + 62.1·11-s − 35.3·13-s − 12.3·14-s + 47.1·16-s − 42.4·17-s − 55.1·19-s + 87.6·20-s + 52.8·22-s − 40.9·23-s + 20.2·25-s − 30.0·26-s + 105.·28-s + 223.·29-s + 107.·31-s + 144.·32-s − 36.0·34-s + 175.·35-s − 7.24·37-s − 46.9·38-s + 156.·40-s − 102.·41-s + ⋯ |
| L(s) = 1 | + 0.300·2-s − 0.909·4-s − 1.07·5-s − 0.784·7-s − 0.574·8-s − 0.324·10-s + 1.70·11-s − 0.753·13-s − 0.236·14-s + 0.736·16-s − 0.605·17-s − 0.666·19-s + 0.980·20-s + 0.511·22-s − 0.371·23-s + 0.161·25-s − 0.226·26-s + 0.713·28-s + 1.42·29-s + 0.624·31-s + 0.795·32-s − 0.182·34-s + 0.845·35-s − 0.0322·37-s − 0.200·38-s + 0.619·40-s − 0.390·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 - 0.850T + 8T^{2} \) |
| 5 | \( 1 + 12.0T + 125T^{2} \) |
| 7 | \( 1 + 14.5T + 343T^{2} \) |
| 11 | \( 1 - 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 42.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 55.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 107.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 7.24T + 5.06e4T^{2} \) |
| 41 | \( 1 + 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 246.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 646.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 180.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 104.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 801.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 944.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 585.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 221.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 544.T + 9.12e5T^{2} \) |
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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
−8.557170968107044241597217015757, −7.55703786589742489236046021749, −6.67668042417016749044917401070, −6.09035624507727913459121064750, −4.84295681466658853936641468610, −4.10470242615947552861378608629, −3.73882788922123476307016123560, −2.59346360727505451109035087821, −0.921476002060106928574871166089, 0,
0.921476002060106928574871166089, 2.59346360727505451109035087821, 3.73882788922123476307016123560, 4.10470242615947552861378608629, 4.84295681466658853936641468610, 6.09035624507727913459121064750, 6.67668042417016749044917401070, 7.55703786589742489236046021749, 8.557170968107044241597217015757
