| L(s) = 1 | − 2.47·2-s + 4.12·4-s − 4.78·7-s − 5.24·8-s + 3.12·11-s + 2.81·13-s + 11.8·14-s + 4.74·16-s + 6.37·17-s + 0.114·19-s − 7.72·22-s + 5.62·23-s − 6.97·26-s − 19.7·28-s − 2.77·29-s − 6.67·31-s − 1.23·32-s − 15.7·34-s − 37-s − 0.282·38-s + 3.12·41-s + 8.57·43-s + 12.8·44-s − 13.9·46-s + 3.40·47-s + 15.8·49-s + 11.6·52-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 2.06·4-s − 1.80·7-s − 1.85·8-s + 0.941·11-s + 0.781·13-s + 3.16·14-s + 1.18·16-s + 1.54·17-s + 0.0262·19-s − 1.64·22-s + 1.17·23-s − 1.36·26-s − 3.72·28-s − 0.515·29-s − 1.19·31-s − 0.218·32-s − 2.70·34-s − 0.164·37-s − 0.0458·38-s + 0.487·41-s + 1.30·43-s + 1.93·44-s − 2.05·46-s + 0.496·47-s + 2.26·49-s + 1.61·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7681414363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7681414363\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 3.12T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 - 0.114T + 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 + 2.77T + 29T^{2} \) |
| 31 | \( 1 + 6.67T + 31T^{2} \) |
| 41 | \( 1 - 3.12T + 41T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 - 3.40T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 9.11T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 4.33T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.68T + 83T^{2} \) |
| 89 | \( 1 - 9.95T + 89T^{2} \) |
| 97 | \( 1 - 5.62T + 97T^{2} \) |
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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
−7.73873548922659186016649514313, −7.33163926909338671653519427219, −6.53086828335186978127955642951, −6.20413814349724944878409074366, −5.36210143985551087214024300650, −3.78911580104066243576489038649, −3.38270617191175824565803211898, −2.47685604909557032172630125083, −1.31044031176914459216539482029, −0.63030890923431822889371315474,
0.63030890923431822889371315474, 1.31044031176914459216539482029, 2.47685604909557032172630125083, 3.38270617191175824565803211898, 3.78911580104066243576489038649, 5.36210143985551087214024300650, 6.20413814349724944878409074366, 6.53086828335186978127955642951, 7.33163926909338671653519427219, 7.73873548922659186016649514313
