| L(s) = 1 | + 5·2-s + 17·4-s + 7·5-s − 13·7-s + 45·8-s + 35·10-s + 26·11-s + 13·13-s − 65·14-s + 89·16-s − 77·17-s − 126·19-s + 119·20-s + 130·22-s + 96·23-s − 76·25-s + 65·26-s − 221·28-s + 82·29-s + 196·31-s + 85·32-s − 385·34-s − 91·35-s − 131·37-s − 630·38-s + 315·40-s − 336·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s + 0.626·5-s − 0.701·7-s + 1.98·8-s + 1.10·10-s + 0.712·11-s + 0.277·13-s − 1.24·14-s + 1.39·16-s − 1.09·17-s − 1.52·19-s + 1.33·20-s + 1.25·22-s + 0.870·23-s − 0.607·25-s + 0.490·26-s − 1.49·28-s + 0.525·29-s + 1.13·31-s + 0.469·32-s − 1.94·34-s − 0.439·35-s − 0.582·37-s − 2.68·38-s + 1.24·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.320261108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.320261108\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 + 13 T + p^{3} T^{2} \) |
| 11 | \( 1 - 26 T + p^{3} T^{2} \) |
| 17 | \( 1 + 77 T + p^{3} T^{2} \) |
| 19 | \( 1 + 126 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 131 T + p^{3} T^{2} \) |
| 41 | \( 1 + 336 T + p^{3} T^{2} \) |
| 43 | \( 1 + 201 T + p^{3} T^{2} \) |
| 47 | \( 1 - 105 T + p^{3} T^{2} \) |
| 53 | \( 1 - 432 T + p^{3} T^{2} \) |
| 59 | \( 1 - 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 56 T + p^{3} T^{2} \) |
| 67 | \( 1 - 478 T + p^{3} T^{2} \) |
| 71 | \( 1 + 9 T + p^{3} T^{2} \) |
| 73 | \( 1 - 98 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1304 T + p^{3} T^{2} \) |
| 83 | \( 1 - 308 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 97 | \( 1 - 70 T + p^{3} T^{2} \) |
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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
−13.28568694628380658107812287449, −12.35744193588294911757248651855, −11.34279658260351786861299167717, −10.22152260392871198293799913832, −8.768731158539995736107990862891, −6.69556606679500742390845941865, −6.29642399189870180572956793355, −4.86238588305200813212080991659, −3.66892350698723826927830135096, −2.21653160079952390520378507380,
2.21653160079952390520378507380, 3.66892350698723826927830135096, 4.86238588305200813212080991659, 6.29642399189870180572956793355, 6.69556606679500742390845941865, 8.768731158539995736107990862891, 10.22152260392871198293799913832, 11.34279658260351786861299167717, 12.35744193588294911757248651855, 13.28568694628380658107812287449
