| L(s) = 1 | − 2.29·3-s − 3.85·5-s + 1.95·7-s + 2.26·9-s − 0.260·11-s − 6.61·13-s + 8.83·15-s + 17-s + 8.40·19-s − 4.47·21-s + 4.64·23-s + 9.82·25-s + 1.69·27-s − 5.50·29-s − 0.591·31-s + 0.596·33-s − 7.50·35-s − 4.47·37-s + 15.1·39-s + 1.62·41-s − 4.97·43-s − 8.70·45-s − 5.88·47-s − 3.19·49-s − 2.29·51-s − 7.20·53-s + 1.00·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 1.72·5-s + 0.737·7-s + 0.753·9-s − 0.0784·11-s − 1.83·13-s + 2.28·15-s + 0.242·17-s + 1.92·19-s − 0.976·21-s + 0.969·23-s + 1.96·25-s + 0.326·27-s − 1.02·29-s − 0.106·31-s + 0.103·33-s − 1.26·35-s − 0.735·37-s + 2.43·39-s + 0.253·41-s − 0.758·43-s − 1.29·45-s − 0.858·47-s − 0.456·49-s − 0.321·51-s − 0.989·53-s + 0.135·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4259587763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4259587763\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 + 0.260T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 19 | \( 1 - 8.40T + 19T^{2} \) |
| 23 | \( 1 - 4.64T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 0.591T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 - 1.62T + 41T^{2} \) |
| 43 | \( 1 + 4.97T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 + 7.20T + 53T^{2} \) |
| 61 | \( 1 + 7.00T + 61T^{2} \) |
| 67 | \( 1 - 0.409T + 67T^{2} \) |
| 71 | \( 1 + 5.14T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 - 0.185T + 89T^{2} \) |
| 97 | \( 1 + 2.56T + 97T^{2} \) |
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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
−7.69252172267317798130428435747, −7.22002328660628239305226839468, −6.64672187336770817088263662460, −5.36037891384571524235431281600, −5.07924338732798420381540522899, −4.61533024385944968111775525779, −3.58405373078217434720961878257, −2.87292384894601635424030009368, −1.40412956960433493410520124756, −0.36967515265028530560432590327,
0.36967515265028530560432590327, 1.40412956960433493410520124756, 2.87292384894601635424030009368, 3.58405373078217434720961878257, 4.61533024385944968111775525779, 5.07924338732798420381540522899, 5.36037891384571524235431281600, 6.64672187336770817088263662460, 7.22002328660628239305226839468, 7.69252172267317798130428435747
