| L(s) = 1 | − 5·5-s + 11.6·7-s + 47.4·11-s + 84.2·13-s + 26.9·17-s − 152.·19-s + 177.·23-s + 25·25-s − 60.0·29-s − 92.1·31-s − 58.1·35-s − 221.·37-s − 115.·41-s + 383.·43-s − 317.·47-s − 207.·49-s + 257.·53-s − 237.·55-s + 642.·59-s − 662.·61-s − 421.·65-s + 597.·67-s + 500.·71-s + 989.·73-s + 552.·77-s − 517.·79-s + 605.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.627·7-s + 1.30·11-s + 1.79·13-s + 0.384·17-s − 1.83·19-s + 1.60·23-s + 0.200·25-s − 0.384·29-s − 0.533·31-s − 0.280·35-s − 0.985·37-s − 0.438·41-s + 1.36·43-s − 0.984·47-s − 0.605·49-s + 0.667·53-s − 0.582·55-s + 1.41·59-s − 1.38·61-s − 0.804·65-s + 1.08·67-s + 0.836·71-s + 1.58·73-s + 0.817·77-s − 0.737·79-s + 0.801·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.510533753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.510533753\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 11.6T + 343T^{2} \) |
| 11 | \( 1 - 47.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 26.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 60.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 92.1T + 2.97e4T^{2} \) |
| 37 | \( 1 + 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 115.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 317.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 662.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 597.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 500.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 989.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 517.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 605.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 742.T + 9.12e5T^{2} \) |
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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
−9.193347875208557265963452813548, −8.719088929015210535055602692772, −8.014376652796992595241418234995, −6.84685857118062351138210845052, −6.28647252428840953667154065897, −5.13599238330691984730084021693, −4.06583029608018745390308098173, −3.48363725633450228573488404023, −1.85088107026802371524457547911, −0.890618167532417611202929288593,
0.890618167532417611202929288593, 1.85088107026802371524457547911, 3.48363725633450228573488404023, 4.06583029608018745390308098173, 5.13599238330691984730084021693, 6.28647252428840953667154065897, 6.84685857118062351138210845052, 8.014376652796992595241418234995, 8.719088929015210535055602692772, 9.193347875208557265963452813548
