| L(s) = 1 | + 2·2-s − 9.10·3-s + 4·4-s − 18.2·6-s + 3.40·7-s + 8·8-s + 55.8·9-s + 61.6·11-s − 36.4·12-s + 77.9·13-s + 6.80·14-s + 16·16-s + 14.8·17-s + 111.·18-s − 46.6·19-s − 30.9·21-s + 123.·22-s + 23·23-s − 72.8·24-s + 155.·26-s − 263.·27-s + 13.6·28-s − 200.·29-s − 20.9·31-s + 32·32-s − 560.·33-s + 29.6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s − 1.23·6-s + 0.183·7-s + 0.353·8-s + 2.07·9-s + 1.68·11-s − 0.876·12-s + 1.66·13-s + 0.129·14-s + 0.250·16-s + 0.211·17-s + 1.46·18-s − 0.563·19-s − 0.321·21-s + 1.19·22-s + 0.208·23-s − 0.619·24-s + 1.17·26-s − 1.87·27-s + 0.0918·28-s − 1.28·29-s − 0.121·31-s + 0.176·32-s − 2.95·33-s + 0.149·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.370469690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370469690\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 9.10T + 27T^{2} \) |
| 7 | \( 1 - 3.40T + 343T^{2} \) |
| 11 | \( 1 - 61.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 14.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 46.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 20.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 233.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 309.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 399.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 190.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 516.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 906.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 33.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 61.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 79.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 533.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 766.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 57.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−9.599991255325861173475859970642, −8.653496987410158952724116584655, −7.32302335704328181423162109081, −6.50875270463853216331530127802, −6.04635820382431183624415493369, −5.32666378085719484387924253345, −4.20276352954904391740215593259, −3.72750985126257802483614459056, −1.68588968297252082859767411775, −0.855889805084134222939989351755,
0.855889805084134222939989351755, 1.68588968297252082859767411775, 3.72750985126257802483614459056, 4.20276352954904391740215593259, 5.32666378085719484387924253345, 6.04635820382431183624415493369, 6.50875270463853216331530127802, 7.32302335704328181423162109081, 8.653496987410158952724116584655, 9.599991255325861173475859970642
