| L(s) = 1 | − 3-s + 5-s − 4.64·7-s + 9-s − 6.12·11-s + 3.47·13-s − 15-s − 6.44·17-s + 4.64·21-s + 6.44·23-s + 25-s − 27-s + 8.96·29-s + 31-s + 6.12·33-s − 4.64·35-s − 2.64·37-s − 3.47·39-s + 8.12·41-s + 6.12·43-s + 45-s + 0.853·47-s + 14.6·49-s + 6.44·51-s − 1.14·53-s − 6.12·55-s − 10.5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.75·7-s + 0.333·9-s − 1.84·11-s + 0.964·13-s − 0.258·15-s − 1.56·17-s + 1.01·21-s + 1.34·23-s + 0.200·25-s − 0.192·27-s + 1.66·29-s + 0.179·31-s + 1.06·33-s − 0.785·35-s − 0.435·37-s − 0.556·39-s + 1.26·41-s + 0.934·43-s + 0.149·45-s + 0.124·47-s + 2.08·49-s + 0.902·51-s − 0.157·53-s − 0.826·55-s − 1.37·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9026283345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9026283345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 4.64T + 7T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 37 | \( 1 + 2.64T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 - 0.853T + 47T^{2} \) |
| 53 | \( 1 + 1.14T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 - 6.11T + 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 + 5.60T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−9.254260181249539032194589033581, −8.650452649905598158600310637576, −7.49995263426666289379004816584, −6.59921874752832423259195253110, −6.19612395598375788745882784606, −5.30833943616629969371929517164, −4.40739500982061174724552559875, −3.13203359390914837788505047810, −2.47207354877037958000623517993, −0.63128021278243154715269417238,
0.63128021278243154715269417238, 2.47207354877037958000623517993, 3.13203359390914837788505047810, 4.40739500982061174724552559875, 5.30833943616629969371929517164, 6.19612395598375788745882784606, 6.59921874752832423259195253110, 7.49995263426666289379004816584, 8.650452649905598158600310637576, 9.254260181249539032194589033581
