| L(s) = 1 | + 1.41·5-s + 3.16·7-s − 4.47·11-s − 4.47·13-s + 6.32·17-s + 2.82·19-s + 4·23-s − 2.99·25-s − 4.24·29-s − 3.16·31-s + 4.47·35-s − 4.47·37-s + 6.32·41-s + 8.48·43-s + 12·47-s + 3.00·49-s + 7.07·53-s − 6.32·55-s + 13.4·61-s − 6.32·65-s + 8·71-s − 4·73-s − 14.1·77-s + 3.16·79-s + 4.47·83-s + 8.94·85-s − 14.1·91-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 1.19·7-s − 1.34·11-s − 1.24·13-s + 1.53·17-s + 0.648·19-s + 0.834·23-s − 0.599·25-s − 0.787·29-s − 0.567·31-s + 0.755·35-s − 0.735·37-s + 0.987·41-s + 1.29·43-s + 1.75·47-s + 0.428·49-s + 0.971·53-s − 0.852·55-s + 1.71·61-s − 0.784·65-s + 0.949·71-s − 0.468·73-s − 1.61·77-s + 0.355·79-s + 0.490·83-s + 0.970·85-s − 1.48·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.369357488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.369357488\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 - 4.47T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.137200477972337339949310560506, −7.48350978492387948199398199798, −7.29159357907075487355288826372, −5.72810406169061293069298865999, −5.42050118480344015855886739654, −4.88873237705608651752637857316, −3.77834763442584886276964835558, −2.65370257639411034436113204450, −2.05861198590112527805779050981, −0.868316741748572091798173712139,
0.868316741748572091798173712139, 2.05861198590112527805779050981, 2.65370257639411034436113204450, 3.77834763442584886276964835558, 4.88873237705608651752637857316, 5.42050118480344015855886739654, 5.72810406169061293069298865999, 7.29159357907075487355288826372, 7.48350978492387948199398199798, 8.137200477972337339949310560506
