| L(s) = 1 | + 1.33·2-s + 3-s − 0.209·4-s + 1.33·6-s − 3.78·7-s − 2.95·8-s + 9-s − 0.209·12-s + 4.93·13-s − 5.06·14-s − 3.53·16-s + 3.25·17-s + 1.33·18-s − 1.37·19-s − 3.78·21-s + 1.93·23-s − 2.95·24-s + 6.60·26-s + 27-s + 0.790·28-s + 1.73·29-s − 5.92·31-s + 1.17·32-s + 4.36·34-s − 0.209·36-s − 4.69·37-s − 1.83·38-s + ⋯ |
| L(s) = 1 | + 0.946·2-s + 0.577·3-s − 0.104·4-s + 0.546·6-s − 1.42·7-s − 1.04·8-s + 0.333·9-s − 0.0603·12-s + 1.36·13-s − 1.35·14-s − 0.884·16-s + 0.790·17-s + 0.315·18-s − 0.314·19-s − 0.825·21-s + 0.403·23-s − 0.603·24-s + 1.29·26-s + 0.192·27-s + 0.149·28-s + 0.323·29-s − 1.06·31-s + 0.208·32-s + 0.747·34-s − 0.0348·36-s − 0.772·37-s − 0.298·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.33T + 2T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 17 | \( 1 - 3.25T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 + 9.43T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 0.805T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 8.59T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 + 2.73T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 3.41T + 83T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.19093221803119854505197906654, −6.54158174067129281454500227555, −5.95384386756828488698452137904, −5.38520148406695710152936422409, −4.40756419862776708429556236097, −3.63603918110538956297332205068, −3.36315584209394516025216254609, −2.63627930257226483940601355060, −1.33334431396348019289872616618, 0,
1.33334431396348019289872616618, 2.63627930257226483940601355060, 3.36315584209394516025216254609, 3.63603918110538956297332205068, 4.40756419862776708429556236097, 5.38520148406695710152936422409, 5.95384386756828488698452137904, 6.54158174067129281454500227555, 7.19093221803119854505197906654
