| L(s) = 1 | − 0.0251·2-s − 6.38·3-s − 7.99·4-s − 5·5-s + 0.160·6-s − 27.7·7-s + 0.403·8-s + 13.7·9-s + 0.125·10-s − 11·11-s + 51.0·12-s + 9.04·13-s + 0.700·14-s + 31.9·15-s + 63.9·16-s − 85.8·17-s − 0.346·18-s + 19·19-s + 39.9·20-s + 177.·21-s + 0.277·22-s + 83.1·23-s − 2.57·24-s + 25·25-s − 0.228·26-s + 84.6·27-s + 222.·28-s + ⋯ |
| L(s) = 1 | − 0.00890·2-s − 1.22·3-s − 0.999·4-s − 0.447·5-s + 0.0109·6-s − 1.50·7-s + 0.0178·8-s + 0.508·9-s + 0.00398·10-s − 0.301·11-s + 1.22·12-s + 0.193·13-s + 0.0133·14-s + 0.549·15-s + 0.999·16-s − 1.22·17-s − 0.00453·18-s + 0.229·19-s + 0.447·20-s + 1.84·21-s + 0.00268·22-s + 0.754·23-s − 0.0218·24-s + 0.200·25-s − 0.00171·26-s + 0.603·27-s + 1.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
| good | 2 | \( 1 + 0.0251T + 8T^{2} \) |
| 3 | \( 1 + 6.38T + 27T^{2} \) |
| 7 | \( 1 + 27.7T + 343T^{2} \) |
| 13 | \( 1 - 9.04T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.8T + 4.91e3T^{2} \) |
| 23 | \( 1 - 83.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 47.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 23.8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 286.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 264.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 472.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 435.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 431.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 258.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 225.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 572.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 246.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 40.2T + 4.93e5T^{2} \) |
| 83 | \( 1 + 313.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−9.217691271974006647662468699363, −8.467437622198701885009056924598, −7.24615191099742002246215746936, −6.43051065906534958104026965640, −5.70616637622554877435220468941, −4.80341660395913253591204301337, −3.94710168156300920227713360690, −2.88498450533724443919183341260, −0.75726354629091154682026213820, 0,
0.75726354629091154682026213820, 2.88498450533724443919183341260, 3.94710168156300920227713360690, 4.80341660395913253591204301337, 5.70616637622554877435220468941, 6.43051065906534958104026965640, 7.24615191099742002246215746936, 8.467437622198701885009056924598, 9.217691271974006647662468699363
