| L(s) = 1 | − 10.2·3-s + 3.08·5-s + 7.31·7-s + 77.6·9-s − 23.9·11-s + 35.5·13-s − 31.5·15-s − 34.2·19-s − 74.7·21-s + 149.·23-s − 115.·25-s − 517.·27-s + 120.·29-s − 247.·31-s + 244.·33-s + 22.5·35-s + 448.·37-s − 363.·39-s − 303.·41-s − 194.·43-s + 239.·45-s + 21.0·47-s − 289.·49-s − 362.·53-s − 73.8·55-s + 350.·57-s − 364.·59-s + ⋯ |
| L(s) = 1 | − 1.96·3-s + 0.275·5-s + 0.394·7-s + 2.87·9-s − 0.656·11-s + 0.758·13-s − 0.542·15-s − 0.413·19-s − 0.777·21-s + 1.35·23-s − 0.923·25-s − 3.69·27-s + 0.769·29-s − 1.43·31-s + 1.29·33-s + 0.108·35-s + 1.99·37-s − 1.49·39-s − 1.15·41-s − 0.688·43-s + 0.792·45-s + 0.0654·47-s − 0.844·49-s − 0.939·53-s − 0.180·55-s + 0.814·57-s − 0.803·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 10.2T + 27T^{2} \) |
| 5 | \( 1 - 3.08T + 125T^{2} \) |
| 7 | \( 1 - 7.31T + 343T^{2} \) |
| 11 | \( 1 + 23.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 247.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 448.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 303.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 194.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 21.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 364.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 478.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 5.17T + 3.00e5T^{2} \) |
| 71 | \( 1 + 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 561.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 746.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 247.T + 9.12e5T^{2} \) |
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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
−8.057798393972684631631561622275, −7.33021165564037907457045538724, −6.39431990226010694944719685365, −6.00424560864706717093587743115, −5.05555478507478748670977298189, −4.69132820919167356236602413173, −3.52747976785023278021337226301, −1.92292193725466961237518354812, −1.02572449286469230881666387527, 0,
1.02572449286469230881666387527, 1.92292193725466961237518354812, 3.52747976785023278021337226301, 4.69132820919167356236602413173, 5.05555478507478748670977298189, 6.00424560864706717093587743115, 6.39431990226010694944719685365, 7.33021165564037907457045538724, 8.057798393972684631631561622275
