| L(s) = 1 | + 2-s + 4-s − 4.24·5-s + 0.296·7-s + 8-s − 4.24·10-s − 2.39·11-s + 2.51·13-s + 0.296·14-s + 16-s − 4.50·17-s − 1.13·19-s − 4.24·20-s − 2.39·22-s + 13.0·25-s + 2.51·26-s + 0.296·28-s + 0.957·29-s + 8.24·31-s + 32-s − 4.50·34-s − 1.25·35-s + 8.34·37-s − 1.13·38-s − 4.24·40-s + 4.10·41-s + 3.86·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.89·5-s + 0.111·7-s + 0.353·8-s − 1.34·10-s − 0.722·11-s + 0.697·13-s + 0.0791·14-s + 0.250·16-s − 1.09·17-s − 0.260·19-s − 0.949·20-s − 0.511·22-s + 2.60·25-s + 0.492·26-s + 0.0559·28-s + 0.177·29-s + 1.48·31-s + 0.176·32-s − 0.773·34-s − 0.212·35-s + 1.37·37-s − 0.184·38-s − 0.671·40-s + 0.641·41-s + 0.589·43-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 - 0.296T + 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 29 | \( 1 - 0.957T + 29T^{2} \) |
| 31 | \( 1 - 8.24T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 - 4.10T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 - 3.35T + 47T^{2} \) |
| 53 | \( 1 - 4.84T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 7.76T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 0.795T + 97T^{2} \) |
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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
−7.49714824049388589748110309127, −6.61244197269333802288246493526, −6.08279441068463681947695802871, −5.02569442250985321290703712720, −4.30381678090705170050378148391, −4.14509858024012777381609834056, −3.07718020698802871867890216030, −2.58456121838927127759869746276, −1.13223127668239435689461192071, 0,
1.13223127668239435689461192071, 2.58456121838927127759869746276, 3.07718020698802871867890216030, 4.14509858024012777381609834056, 4.30381678090705170050378148391, 5.02569442250985321290703712720, 6.08279441068463681947695802871, 6.61244197269333802288246493526, 7.49714824049388589748110309127
