| L(s) = 1 | + 4.12·2-s + 9·4-s − 13.4·5-s − 31.4·7-s + 4.12·8-s − 55.4·10-s − 40.4·11-s − 129.·14-s − 55·16-s + 43.1·17-s − 26.9·19-s − 120.·20-s − 166.·22-s + 19.0·23-s + 55.6·25-s − 282.·28-s + 154.·29-s + 308.·31-s − 259.·32-s + 177.·34-s + 422.·35-s + 43.5·37-s − 111.·38-s − 55.4·40-s − 47.8·41-s + 342.·43-s − 364.·44-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.12·4-s − 1.20·5-s − 1.69·7-s + 0.182·8-s − 1.75·10-s − 1.11·11-s − 2.47·14-s − 0.859·16-s + 0.615·17-s − 0.325·19-s − 1.35·20-s − 1.61·22-s + 0.172·23-s + 0.445·25-s − 1.90·28-s + 0.986·29-s + 1.78·31-s − 1.43·32-s + 0.897·34-s + 2.03·35-s + 0.193·37-s − 0.474·38-s − 0.219·40-s − 0.182·41-s + 1.21·43-s − 1.24·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.676847286\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676847286\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.12T + 8T^{2} \) |
| 5 | \( 1 + 13.4T + 125T^{2} \) |
| 7 | \( 1 + 31.4T + 343T^{2} \) |
| 11 | \( 1 + 40.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 43.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 19.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 43.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 47.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 342.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 590.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 541.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 230.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 897.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.56e3T + 9.12e5T^{2} \) |
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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
−9.089460413523030163258474690883, −8.086318185734075828161284073037, −7.27683759757721520470902140810, −6.42639117531139519019351153973, −5.80492455606930698474407119857, −4.72550328205543154914964092093, −4.04672663918203774942455230607, −3.09597516843740866887606803823, −2.76367935986095977221814498973, −0.47348700764199541273479608949,
0.47348700764199541273479608949, 2.76367935986095977221814498973, 3.09597516843740866887606803823, 4.04672663918203774942455230607, 4.72550328205543154914964092093, 5.80492455606930698474407119857, 6.42639117531139519019351153973, 7.27683759757721520470902140810, 8.086318185734075828161284073037, 9.089460413523030163258474690883
