| L(s) = 1 | + 2.40·2-s − 2.20·4-s + 6.97·5-s + 19.5·7-s − 24.5·8-s + 16.7·10-s + 7.19·11-s + 72.3·13-s + 47.0·14-s − 41.5·16-s − 52.4·17-s − 91.4·19-s − 15.3·20-s + 17.3·22-s − 123.·23-s − 76.4·25-s + 174.·26-s − 43.0·28-s − 185.·29-s − 290.·31-s + 96.5·32-s − 126.·34-s + 136.·35-s + 33.4·37-s − 220.·38-s − 171.·40-s + 206.·41-s + ⋯ |
| L(s) = 1 | + 0.851·2-s − 0.275·4-s + 0.623·5-s + 1.05·7-s − 1.08·8-s + 0.530·10-s + 0.197·11-s + 1.54·13-s + 0.897·14-s − 0.648·16-s − 0.748·17-s − 1.10·19-s − 0.171·20-s + 0.167·22-s − 1.12·23-s − 0.611·25-s + 1.31·26-s − 0.290·28-s − 1.18·29-s − 1.68·31-s + 0.533·32-s − 0.637·34-s + 0.657·35-s + 0.148·37-s − 0.939·38-s − 0.676·40-s + 0.788·41-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 | \( 1 - 2.40T + 8T^{2} \) |
| 5 | \( 1 - 6.97T + 125T^{2} \) |
| 7 | \( 1 - 19.5T + 343T^{2} \) |
| 11 | \( 1 - 7.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 290.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 33.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 206.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 156.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 84.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 374.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 649.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 978.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 285.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.06e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 663.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.39e3T + 9.12e5T^{2} \) |
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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
−8.494243689326187557399463651744, −7.60274141809009832396772302578, −6.33441110144633723669940171791, −5.90071691736758056623455761868, −5.16307595266328557285288611955, −4.09542164878679840752649468603, −3.81661269045478850207853218463, −2.29852682873260950795358865398, −1.53557639890945988648570510215, 0,
1.53557639890945988648570510215, 2.29852682873260950795358865398, 3.81661269045478850207853218463, 4.09542164878679840752649468603, 5.16307595266328557285288611955, 5.90071691736758056623455761868, 6.33441110144633723669940171791, 7.60274141809009832396772302578, 8.494243689326187557399463651744
