| L(s) = 1 | − 4.88·3-s + 8.53·5-s − 34.6·7-s − 3.11·9-s − 39.6·11-s − 21.0·13-s − 41.6·15-s − 16.3·17-s + 65.9·19-s + 169.·21-s − 7.29·23-s − 52.2·25-s + 147.·27-s + 115.·29-s − 131.·31-s + 193.·33-s − 295.·35-s − 37·37-s + 103.·39-s − 214.·41-s − 181.·43-s − 26.5·45-s + 254.·47-s + 859.·49-s + 79.8·51-s + 143.·53-s − 338.·55-s + ⋯ |
| L(s) = 1 | − 0.940·3-s + 0.763·5-s − 1.87·7-s − 0.115·9-s − 1.08·11-s − 0.450·13-s − 0.717·15-s − 0.233·17-s + 0.796·19-s + 1.76·21-s − 0.0661·23-s − 0.417·25-s + 1.04·27-s + 0.739·29-s − 0.760·31-s + 1.02·33-s − 1.42·35-s − 0.164·37-s + 0.423·39-s − 0.818·41-s − 0.644·43-s − 0.0881·45-s + 0.789·47-s + 2.50·49-s + 0.219·51-s + 0.371·53-s − 0.828·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6176131427\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6176131427\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 + 4.88T + 27T^{2} \) |
| 5 | \( 1 - 8.53T + 125T^{2} \) |
| 7 | \( 1 + 34.6T + 343T^{2} \) |
| 11 | \( 1 + 39.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 21.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 7.29T + 1.21e4T^{2} \) |
| 29 | \( 1 - 115.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 131.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 214.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 254.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 143.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 74.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 865.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 382.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 851.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 675.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 460.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 205.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.01e3T + 9.12e5T^{2} \) |
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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
−10.08751632666213139124914153455, −9.791156015048968745008890758405, −8.703193799135516821183979180837, −7.31775450192692084456334894175, −6.46130045547623972746566736073, −5.76420677978807472872221903425, −5.05782507107673569939152693323, −3.40475615383566480230292880990, −2.42860570820703666185620083063, −0.46081808176369600382673487628,
0.46081808176369600382673487628, 2.42860570820703666185620083063, 3.40475615383566480230292880990, 5.05782507107673569939152693323, 5.76420677978807472872221903425, 6.46130045547623972746566736073, 7.31775450192692084456334894175, 8.703193799135516821183979180837, 9.791156015048968745008890758405, 10.08751632666213139124914153455
