| L(s) = 1 | + 1.73·3-s + 2.58·5-s − 2.10·7-s + 0.0181·9-s + 2.62·11-s + 0.715·13-s + 4.49·15-s − 5.38·17-s − 1.38·19-s − 3.65·21-s − 6.72·23-s + 1.68·25-s − 5.18·27-s − 4.19·29-s + 5.92·31-s + 4.55·33-s − 5.43·35-s − 7.49·37-s + 1.24·39-s − 2.29·41-s − 7.41·43-s + 0.0468·45-s + 7.28·47-s − 2.57·49-s − 9.34·51-s − 11.8·53-s + 6.78·55-s + ⋯ |
| L(s) = 1 | + 1.00·3-s + 1.15·5-s − 0.794·7-s + 0.00604·9-s + 0.790·11-s + 0.198·13-s + 1.16·15-s − 1.30·17-s − 0.317·19-s − 0.797·21-s − 1.40·23-s + 0.337·25-s − 0.996·27-s − 0.778·29-s + 1.06·31-s + 0.793·33-s − 0.919·35-s − 1.23·37-s + 0.198·39-s − 0.357·41-s − 1.13·43-s + 0.00699·45-s + 1.06·47-s − 0.368·49-s − 1.30·51-s − 1.63·53-s + 0.914·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.73T + 3T^{2} \) |
| 5 | \( 1 - 2.58T + 5T^{2} \) |
| 7 | \( 1 + 2.10T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 - 0.715T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 + 1.38T + 19T^{2} \) |
| 23 | \( 1 + 6.72T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 + 2.29T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 7.39T + 59T^{2} \) |
| 61 | \( 1 + 1.47T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 + 9.67T + 73T^{2} \) |
| 79 | \( 1 + 9.04T + 79T^{2} \) |
| 83 | \( 1 + 4.25T + 83T^{2} \) |
| 89 | \( 1 - 9.05T + 89T^{2} \) |
| 97 | \( 1 - 8.08T + 97T^{2} \) |
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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
−7.51808529524351862914898085363, −6.58307187528995790896515022397, −6.27477884929173377581970568153, −5.58079860761869769829069616930, −4.50973143149963545169717647985, −3.73903630337895332255994765454, −3.05258688878418259498146557073, −2.15214773542203296996723571427, −1.71057175729013531333331852644, 0,
1.71057175729013531333331852644, 2.15214773542203296996723571427, 3.05258688878418259498146557073, 3.73903630337895332255994765454, 4.50973143149963545169717647985, 5.58079860761869769829069616930, 6.27477884929173377581970568153, 6.58307187528995790896515022397, 7.51808529524351862914898085363
