| L(s) = 1 | − 2-s + 4-s − 1.83·5-s − 1.24·7-s − 8-s + 1.83·10-s + 0.102·11-s − 3.87·13-s + 1.24·14-s + 16-s + 3.06·17-s − 4.97·19-s − 1.83·20-s − 0.102·22-s − 1.63·25-s + 3.87·26-s − 1.24·28-s − 6.71·29-s + 3.86·31-s − 32-s − 3.06·34-s + 2.28·35-s + 2.55·37-s + 4.97·38-s + 1.83·40-s − 1.65·41-s − 3.59·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.820·5-s − 0.470·7-s − 0.353·8-s + 0.580·10-s + 0.0308·11-s − 1.07·13-s + 0.332·14-s + 0.250·16-s + 0.742·17-s − 1.14·19-s − 0.410·20-s − 0.0218·22-s − 0.326·25-s + 0.759·26-s − 0.235·28-s − 1.24·29-s + 0.694·31-s − 0.176·32-s − 0.525·34-s + 0.385·35-s + 0.419·37-s + 0.807·38-s + 0.290·40-s − 0.258·41-s − 0.548·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4294914338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4294914338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 - 0.102T + 11T^{2} \) |
| 13 | \( 1 + 3.87T + 13T^{2} \) |
| 17 | \( 1 - 3.06T + 17T^{2} \) |
| 19 | \( 1 + 4.97T + 19T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 3.59T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 67 | \( 1 - 1.55T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 5.20T + 83T^{2} \) |
| 89 | \( 1 - 7.97T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.68660209090002954170802834145, −7.23680830696625644735231943964, −6.47744444362323865435964104682, −5.80960387251723690512107256207, −4.89553884623314068154125712621, −4.11248407993371495344070734682, −3.36108163302974398864820213674, −2.56716858034212223455420489334, −1.64594580709080427646626424439, −0.33977894229454638854255695211,
0.33977894229454638854255695211, 1.64594580709080427646626424439, 2.56716858034212223455420489334, 3.36108163302974398864820213674, 4.11248407993371495344070734682, 4.89553884623314068154125712621, 5.80960387251723690512107256207, 6.47744444362323865435964104682, 7.23680830696625644735231943964, 7.68660209090002954170802834145
