| L(s) = 1 | − 7.11·3-s − 16.3·5-s + 5.74·7-s + 23.6·9-s − 23.3·11-s + 18.0·13-s + 116.·15-s − 24.1·17-s + 12.0·19-s − 40.8·21-s − 144.·23-s + 143.·25-s + 24.1·27-s − 29·29-s + 6.27·31-s + 166.·33-s − 94.1·35-s − 28.6·37-s − 128.·39-s − 436.·41-s − 495.·43-s − 386.·45-s + 351.·47-s − 309.·49-s + 171.·51-s + 58.0·53-s + 382.·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s − 1.46·5-s + 0.310·7-s + 0.874·9-s − 0.640·11-s + 0.384·13-s + 2.00·15-s − 0.344·17-s + 0.145·19-s − 0.424·21-s − 1.31·23-s + 1.14·25-s + 0.171·27-s − 0.185·29-s + 0.0363·31-s + 0.877·33-s − 0.454·35-s − 0.127·37-s − 0.526·39-s − 1.66·41-s − 1.75·43-s − 1.28·45-s + 1.09·47-s − 0.903·49-s + 0.471·51-s + 0.150·53-s + 0.938·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1056681351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1056681351\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 7.11T + 27T^{2} \) |
| 5 | \( 1 + 16.3T + 125T^{2} \) |
| 7 | \( 1 - 5.74T + 343T^{2} \) |
| 11 | \( 1 + 23.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 12.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 144.T + 1.21e4T^{2} \) |
| 31 | \( 1 - 6.27T + 2.97e4T^{2} \) |
| 37 | \( 1 + 28.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 495.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 351.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 58.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 485.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 607.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 296.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 330.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 662.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 145.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 851.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 227.T + 9.12e5T^{2} \) |
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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
−8.603475648753646542366046138828, −8.061982160631986395337535283943, −7.26671058117883564444122140818, −6.48004323398432733285615146530, −5.61034954599762200535161790148, −4.81612947147568494393861384767, −4.12868478991709858899137038348, −3.13306760235205853456441211547, −1.54499332472573280125183218416, −0.16612842489638316736938986530,
0.16612842489638316736938986530, 1.54499332472573280125183218416, 3.13306760235205853456441211547, 4.12868478991709858899137038348, 4.81612947147568494393861384767, 5.61034954599762200535161790148, 6.48004323398432733285615146530, 7.26671058117883564444122140818, 8.061982160631986395337535283943, 8.603475648753646542366046138828
