| L(s) = 1 | − 3-s − 2.96·5-s + 3.35·7-s + 9-s − 1.61·11-s + 13-s + 2.96·15-s + 2·17-s − 3.35·19-s − 3.35·21-s − 6.70·23-s + 3.77·25-s − 27-s + 2·29-s + 6.57·31-s + 1.61·33-s − 9.92·35-s + 7.92·37-s − 39-s + 6.96·41-s − 0.775·43-s − 2.96·45-s + 2.38·47-s + 4.22·49-s − 2·51-s + 11.9·53-s + 4.77·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.32·5-s + 1.26·7-s + 0.333·9-s − 0.486·11-s + 0.277·13-s + 0.764·15-s + 0.485·17-s − 0.768·19-s − 0.731·21-s − 1.39·23-s + 0.755·25-s − 0.192·27-s + 0.371·29-s + 1.18·31-s + 0.280·33-s − 1.67·35-s + 1.30·37-s − 0.160·39-s + 1.08·41-s − 0.118·43-s − 0.441·45-s + 0.348·47-s + 0.603·49-s − 0.280·51-s + 1.63·53-s + 0.643·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.105306086\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.105306086\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 6.57T + 31T^{2} \) |
| 37 | \( 1 - 7.92T + 37T^{2} \) |
| 41 | \( 1 - 6.96T + 41T^{2} \) |
| 43 | \( 1 + 0.775T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 0.312T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 8.12T + 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 - 0.0752T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 8.31T + 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.92T + 97T^{2} \) |
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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
−9.909379571298621646075441494265, −8.517800766796481598436269093152, −8.040938357485383457994899069343, −7.48387630450277437107113140678, −6.35504145816321506991078193235, −5.39155252434833116677710501604, −4.44074000228001111749730067190, −3.91778161693034905165685159112, −2.35333955724718377733505261174, −0.810703796064006188286456993622,
0.810703796064006188286456993622, 2.35333955724718377733505261174, 3.91778161693034905165685159112, 4.44074000228001111749730067190, 5.39155252434833116677710501604, 6.35504145816321506991078193235, 7.48387630450277437107113140678, 8.040938357485383457994899069343, 8.517800766796481598436269093152, 9.909379571298621646075441494265
