| L(s) = 1 | − 4.53·2-s + 12.5·4-s − 5·5-s − 7·7-s − 20.5·8-s + 22.6·10-s + 19.0·11-s − 2.93·13-s + 31.7·14-s − 7.21·16-s + 6.49·17-s − 5.43·19-s − 62.6·20-s − 86.3·22-s − 49.3·23-s + 25·25-s + 13.3·26-s − 87.7·28-s + 291.·29-s + 244.·31-s + 196.·32-s − 29.4·34-s + 35·35-s − 193.·37-s + 24.6·38-s + 102.·40-s − 315.·41-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.56·4-s − 0.447·5-s − 0.377·7-s − 0.907·8-s + 0.716·10-s + 0.522·11-s − 0.0626·13-s + 0.605·14-s − 0.112·16-s + 0.0927·17-s − 0.0656·19-s − 0.700·20-s − 0.837·22-s − 0.447·23-s + 0.200·25-s + 0.100·26-s − 0.592·28-s + 1.86·29-s + 1.41·31-s + 1.08·32-s − 0.148·34-s + 0.169·35-s − 0.858·37-s + 0.105·38-s + 0.405·40-s − 1.20·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 4.53T + 8T^{2} \) |
| 11 | \( 1 - 19.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.93T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.49T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.43T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 86.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 509.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 83.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 5.25T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 863.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 326.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−10.34301194838350969192057189641, −9.912917910576036660002778366986, −8.733023208634280286081840199193, −8.220907487845438659299340479149, −7.08460507343796884953169348500, −6.34244154936345227806499879321, −4.56804878502591952579584650807, −2.99071596766604524419226575158, −1.38628743103439624835248470911, 0,
1.38628743103439624835248470911, 2.99071596766604524419226575158, 4.56804878502591952579584650807, 6.34244154936345227806499879321, 7.08460507343796884953169348500, 8.220907487845438659299340479149, 8.733023208634280286081840199193, 9.912917910576036660002778366986, 10.34301194838350969192057189641
