| L(s) = 1 | + 0.311·2-s − 2.90·3-s − 1.90·4-s − 5-s − 0.903·6-s − 1.21·8-s + 5.42·9-s − 0.311·10-s − 1.52·11-s + 5.52·12-s + 0.622·13-s + 2.90·15-s + 3.42·16-s + 7.95·17-s + 1.68·18-s + 1.09·19-s + 1.90·20-s − 0.474·22-s + 7.52·23-s + 3.52·24-s + 25-s + 0.193·26-s − 7.05·27-s − 29-s + 0.903·30-s + 6.90·31-s + 3.49·32-s + ⋯ |
| L(s) = 1 | + 0.219·2-s − 1.67·3-s − 0.951·4-s − 0.447·5-s − 0.368·6-s − 0.429·8-s + 1.80·9-s − 0.0983·10-s − 0.459·11-s + 1.59·12-s + 0.172·13-s + 0.749·15-s + 0.857·16-s + 1.92·17-s + 0.398·18-s + 0.251·19-s + 0.425·20-s − 0.101·22-s + 1.56·23-s + 0.719·24-s + 0.200·25-s + 0.0379·26-s − 1.35·27-s − 0.185·29-s + 0.164·30-s + 1.23·31-s + 0.617·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8524338106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8524338106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.311T + 2T^{2} \) |
| 3 | \( 1 + 2.90T + 3T^{2} \) |
| 11 | \( 1 + 1.52T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 - 7.95T + 17T^{2} \) |
| 19 | \( 1 - 1.09T + 19T^{2} \) |
| 23 | \( 1 - 7.52T + 23T^{2} \) |
| 31 | \( 1 - 6.90T + 31T^{2} \) |
| 37 | \( 1 - 3.95T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 6.90T + 47T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 - 1.67T + 59T^{2} \) |
| 61 | \( 1 - 1.86T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 9.13T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.945557574579502203729604102492, −7.07347124709586413116758430667, −6.41206504746946424421650534706, −5.54525432744848025520565310558, −5.16951855885689172223373278470, −4.68946100721651651806758754269, −3.72203777906503592039903505028, −3.04212942244126921001894949381, −1.21693149359206007935981309955, −0.60164466764709793396470255615,
0.60164466764709793396470255615, 1.21693149359206007935981309955, 3.04212942244126921001894949381, 3.72203777906503592039903505028, 4.68946100721651651806758754269, 5.16951855885689172223373278470, 5.54525432744848025520565310558, 6.41206504746946424421650534706, 7.07347124709586413116758430667, 7.945557574579502203729604102492
