| L(s) = 1 | + 0.676·3-s − 1.78·5-s + 3.10·7-s − 2.54·9-s − 0.839·11-s − 2.15·13-s − 1.20·15-s + 5.99·17-s + 2.09·21-s − 7.65·23-s − 1.80·25-s − 3.74·27-s + 2.35·29-s − 3.43·31-s − 0.567·33-s − 5.54·35-s − 9.49·37-s − 1.45·39-s + 2.73·41-s + 6.36·43-s + 4.54·45-s − 5.51·47-s + 2.61·49-s + 4.05·51-s + 5.87·53-s + 1.50·55-s + 8.43·59-s + ⋯ |
| L(s) = 1 | + 0.390·3-s − 0.799·5-s + 1.17·7-s − 0.847·9-s − 0.253·11-s − 0.597·13-s − 0.312·15-s + 1.45·17-s + 0.457·21-s − 1.59·23-s − 0.360·25-s − 0.721·27-s + 0.437·29-s − 0.616·31-s − 0.0988·33-s − 0.937·35-s − 1.56·37-s − 0.233·39-s + 0.427·41-s + 0.969·43-s + 0.677·45-s − 0.804·47-s + 0.374·49-s + 0.567·51-s + 0.806·53-s + 0.202·55-s + 1.09·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.676T + 3T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + 0.839T + 11T^{2} \) |
| 13 | \( 1 + 2.15T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 + 9.49T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 - 6.36T + 43T^{2} \) |
| 47 | \( 1 + 5.51T + 47T^{2} \) |
| 53 | \( 1 - 5.87T + 53T^{2} \) |
| 59 | \( 1 - 8.43T + 59T^{2} \) |
| 61 | \( 1 - 8.12T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 4.76T + 79T^{2} \) |
| 83 | \( 1 + 0.304T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−8.284175225183296895262306655084, −7.73742682042458756631812826115, −7.28141104960703994176624812070, −5.85830434323428098472423547154, −5.35934000865903411656979340628, −4.36459552270627534612962042330, −3.60460291303802527785219722958, −2.64203212191837800229751289404, −1.60052418973776043526650792720, 0,
1.60052418973776043526650792720, 2.64203212191837800229751289404, 3.60460291303802527785219722958, 4.36459552270627534612962042330, 5.35934000865903411656979340628, 5.85830434323428098472423547154, 7.28141104960703994176624812070, 7.73742682042458756631812826115, 8.284175225183296895262306655084
