L(s) = 1 | + 2-s + 3-s + 4-s − 2.51·5-s + 6-s + 1.73·7-s + 8-s + 9-s − 2.51·10-s − 0.642·11-s + 12-s − 2.58·13-s + 1.73·14-s − 2.51·15-s + 16-s − 7.71·17-s + 18-s − 19-s − 2.51·20-s + 1.73·21-s − 0.642·22-s + 5.46·23-s + 24-s + 1.30·25-s − 2.58·26-s + 27-s + 1.73·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.12·5-s + 0.408·6-s + 0.656·7-s + 0.353·8-s + 0.333·9-s − 0.794·10-s − 0.193·11-s + 0.288·12-s − 0.716·13-s + 0.464·14-s − 0.648·15-s + 0.250·16-s − 1.87·17-s + 0.235·18-s − 0.229·19-s − 0.561·20-s + 0.379·21-s − 0.136·22-s + 1.13·23-s + 0.204·24-s + 0.261·25-s − 0.506·26-s + 0.192·27-s + 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 + T \) |
good | 5 | \( 1 + 2.51T + 5T^{2} \) |
| 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 0.642T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 7.71T + 17T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 - 8.07T + 29T^{2} \) |
| 31 | \( 1 + 0.677T + 31T^{2} \) |
| 37 | \( 1 + 7.32T + 37T^{2} \) |
| 41 | \( 1 + 6.11T + 41T^{2} \) |
| 43 | \( 1 + 1.89T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 59 | \( 1 - 6.39T + 59T^{2} \) |
| 61 | \( 1 + 4.23T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 5.86T + 73T^{2} \) |
| 79 | \( 1 + 9.67T + 79T^{2} \) |
| 83 | \( 1 + 0.708T + 83T^{2} \) |
| 89 | \( 1 - 9.33T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.76819700533368995792847310648, −6.81996468227292830817424392618, −6.68999907790620966286650380021, −5.16730007318337683867412831371, −4.73837170383946232620988216030, −4.14591264019218556515318035444, −3.26949681925658433248578387754, −2.53963217597067988118620076603, −1.60114781012784567292016207943, 0,
1.60114781012784567292016207943, 2.53963217597067988118620076603, 3.26949681925658433248578387754, 4.14591264019218556515318035444, 4.73837170383946232620988216030, 5.16730007318337683867412831371, 6.68999907790620966286650380021, 6.81996468227292830817424392618, 7.76819700533368995792847310648