| L(s) = 1 | + 4.23·2-s + 9.96·4-s + 4.25·5-s + 23.9·7-s + 8.34·8-s + 18.0·10-s − 15.7·11-s − 48.2·13-s + 101.·14-s − 44.3·16-s − 71.9·17-s − 87.6·19-s + 42.4·20-s − 66.9·22-s + 114.·23-s − 106.·25-s − 204.·26-s + 239.·28-s − 230.·29-s + 67.5·31-s − 254.·32-s − 304.·34-s + 102.·35-s − 118.·37-s − 371.·38-s + 35.5·40-s + 20.2·41-s + ⋯ |
| L(s) = 1 | + 1.49·2-s + 1.24·4-s + 0.381·5-s + 1.29·7-s + 0.368·8-s + 0.571·10-s − 0.432·11-s − 1.03·13-s + 1.94·14-s − 0.693·16-s − 1.02·17-s − 1.05·19-s + 0.474·20-s − 0.648·22-s + 1.03·23-s − 0.854·25-s − 1.54·26-s + 1.61·28-s − 1.47·29-s + 0.391·31-s − 1.40·32-s − 1.53·34-s + 0.493·35-s − 0.524·37-s − 1.58·38-s + 0.140·40-s + 0.0771·41-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 | \( 1 - 4.23T + 8T^{2} \) |
| 5 | \( 1 - 4.25T + 125T^{2} \) |
| 7 | \( 1 - 23.9T + 343T^{2} \) |
| 11 | \( 1 + 15.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 48.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 87.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 230.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 20.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 186.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 535.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 392.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 280.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 230.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 567.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 546.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 245.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 0.989T + 4.93e5T^{2} \) |
| 83 | \( 1 + 475.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 343.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 437.T + 9.12e5T^{2} \) |
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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
−8.234851814986247405127701738301, −7.32619154274806689133902114433, −6.58385358119928731673798234099, −5.66956251109370033827073935900, −4.92464896140558806892111507469, −4.58166629583733108148428783277, −3.54279038511923855054632376867, −2.35190907069849975454863727685, −1.87860130328587842652051848873, 0,
1.87860130328587842652051848873, 2.35190907069849975454863727685, 3.54279038511923855054632376867, 4.58166629583733108148428783277, 4.92464896140558806892111507469, 5.66956251109370033827073935900, 6.58385358119928731673798234099, 7.32619154274806689133902114433, 8.234851814986247405127701738301
