| L(s) = 1 | − 1.16·2-s − 0.640·4-s − 5-s − 0.957·7-s + 3.07·8-s + 1.16·10-s − 5.05·11-s + 1.11·14-s − 2.30·16-s + 1.25·17-s − 2.44·19-s + 0.640·20-s + 5.89·22-s + 5.45·23-s + 25-s + 0.613·28-s − 10.2·29-s + 1.02·31-s − 3.46·32-s − 1.46·34-s + 0.957·35-s + 0.939·37-s + 2.85·38-s − 3.07·40-s − 7.55·41-s + 0.259·43-s + 3.24·44-s + ⋯ |
| L(s) = 1 | − 0.824·2-s − 0.320·4-s − 0.447·5-s − 0.361·7-s + 1.08·8-s + 0.368·10-s − 1.52·11-s + 0.298·14-s − 0.576·16-s + 0.304·17-s − 0.561·19-s + 0.143·20-s + 1.25·22-s + 1.13·23-s + 0.200·25-s + 0.115·28-s − 1.89·29-s + 0.184·31-s − 0.612·32-s − 0.251·34-s + 0.161·35-s + 0.154·37-s + 0.462·38-s − 0.486·40-s − 1.18·41-s + 0.0395·43-s + 0.488·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3590075241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3590075241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 7 | \( 1 + 0.957T + 7T^{2} \) |
| 11 | \( 1 + 5.05T + 11T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 - 0.939T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 - 0.259T + 43T^{2} \) |
| 47 | \( 1 - 0.115T + 47T^{2} \) |
| 53 | \( 1 - 2.43T + 53T^{2} \) |
| 59 | \( 1 + 8.32T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 0.977T + 71T^{2} \) |
| 73 | \( 1 + 7.06T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 3.76T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.044926995366403789792906595694, −7.32808418327837752067761936865, −6.83861942574545838247709957861, −5.61792396766269575693729544639, −5.14672248948522968265023004620, −4.32608191202760152891744836104, −3.49714173641585595512147237568, −2.63963625545304508151874782424, −1.57394950195537778831245232801, −0.34395392825207224167531834622,
0.34395392825207224167531834622, 1.57394950195537778831245232801, 2.63963625545304508151874782424, 3.49714173641585595512147237568, 4.32608191202760152891744836104, 5.14672248948522968265023004620, 5.61792396766269575693729544639, 6.83861942574545838247709957861, 7.32808418327837752067761936865, 8.044926995366403789792906595694
