| L(s) = 1 | − 1.61·2-s − 1.18·3-s + 0.621·4-s + 1.92·6-s − 2.23·7-s + 2.23·8-s − 1.58·9-s + 5.64·11-s − 0.738·12-s + 4.70·13-s + 3.61·14-s − 4.85·16-s + 0.785·17-s + 2.56·18-s + 2.65·21-s − 9.13·22-s + 5.13·23-s − 2.65·24-s − 7.61·26-s + 5.45·27-s − 1.38·28-s − 3.03·29-s − 8.10·31-s + 3.39·32-s − 6.70·33-s − 1.27·34-s − 0.986·36-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.686·3-s + 0.310·4-s + 0.785·6-s − 0.843·7-s + 0.789·8-s − 0.529·9-s + 1.70·11-s − 0.213·12-s + 1.30·13-s + 0.965·14-s − 1.21·16-s + 0.190·17-s + 0.605·18-s + 0.578·21-s − 1.94·22-s + 1.06·23-s − 0.541·24-s − 1.49·26-s + 1.04·27-s − 0.262·28-s − 0.563·29-s − 1.45·31-s + 0.600·32-s − 1.16·33-s − 0.218·34-s − 0.164·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.18T + 3T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 - 5.64T + 11T^{2} \) |
| 13 | \( 1 - 4.70T + 13T^{2} \) |
| 17 | \( 1 - 0.785T + 17T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 + 3.03T + 29T^{2} \) |
| 31 | \( 1 + 8.10T + 31T^{2} \) |
| 37 | \( 1 + 0.985T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 0.960T + 47T^{2} \) |
| 53 | \( 1 - 4.41T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 - 2.09T + 61T^{2} \) |
| 67 | \( 1 + 3.24T + 67T^{2} \) |
| 71 | \( 1 + 7.17T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 1.33T + 79T^{2} \) |
| 83 | \( 1 - 7.52T + 83T^{2} \) |
| 89 | \( 1 + 3.40T + 89T^{2} \) |
| 97 | \( 1 + 12.7T + 97T^{2} \) |
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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
−7.28589290730561437463257800696, −6.82373686773649264871444969434, −6.13093699381295213732093552838, −5.61685996002349439438734008499, −4.57541266525392275110741186247, −3.76950475517477172195931067846, −3.12388214594840982328245148902, −1.67342631885314193433196469565, −1.01948473109136851450408521197, 0,
1.01948473109136851450408521197, 1.67342631885314193433196469565, 3.12388214594840982328245148902, 3.76950475517477172195931067846, 4.57541266525392275110741186247, 5.61685996002349439438734008499, 6.13093699381295213732093552838, 6.82373686773649264871444969434, 7.28589290730561437463257800696
