| L(s) = 1 | + 3.71·2-s + 5.79·4-s + 14.6·5-s − 2.58·7-s − 8.17·8-s + 54.3·10-s + 4.27·11-s − 0.233·13-s − 9.58·14-s − 76.7·16-s + 23.9·17-s + 124.·19-s + 84.7·20-s + 15.8·22-s + 160.·23-s + 88.6·25-s − 0.867·26-s − 14.9·28-s − 262.·29-s + 36.9·31-s − 219.·32-s + 89.0·34-s − 37.7·35-s + 449.·37-s + 461.·38-s − 119.·40-s − 109.·41-s + ⋯ |
| L(s) = 1 | + 1.31·2-s + 0.724·4-s + 1.30·5-s − 0.139·7-s − 0.361·8-s + 1.71·10-s + 0.117·11-s − 0.00498·13-s − 0.183·14-s − 1.19·16-s + 0.341·17-s + 1.50·19-s + 0.947·20-s + 0.153·22-s + 1.45·23-s + 0.709·25-s − 0.00654·26-s − 0.101·28-s − 1.68·29-s + 0.213·31-s − 1.21·32-s + 0.449·34-s − 0.182·35-s + 1.99·37-s + 1.97·38-s − 0.472·40-s − 0.417·41-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
| \(L(2)\) |
\(\approx\) |
\(6.087593124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.087593124\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.71T + 8T^{2} \) |
| 5 | \( 1 - 14.6T + 125T^{2} \) |
| 7 | \( 1 + 2.58T + 343T^{2} \) |
| 11 | \( 1 - 4.27T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.233T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 124.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 262.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 36.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 449.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 109.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 275.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 463.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 62.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 54.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 564.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 812.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 52.2T + 3.57e5T^{2} \) |
| 73 | \( 1 + 375.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 19.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 378.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 190.T + 9.12e5T^{2} \) |
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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
−9.033839827146930888999661257062, −7.72198372711074057174703033872, −6.83691227561396765065913859694, −6.09053978102975403477783142900, −5.40720021642954845582843556329, −4.98172300449649488326684231437, −3.78858652440293569979128520051, −3.01943113953390443732912060032, −2.15196901998633991524905324290, −0.944057121949584398485599323743,
0.944057121949584398485599323743, 2.15196901998633991524905324290, 3.01943113953390443732912060032, 3.78858652440293569979128520051, 4.98172300449649488326684231437, 5.40720021642954845582843556329, 6.09053978102975403477783142900, 6.83691227561396765065913859694, 7.72198372711074057174703033872, 9.033839827146930888999661257062
