| L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·11-s − 13-s − 16-s − 17-s + 8·19-s + 4·22-s − 5·25-s + 26-s − 8·29-s + 8·31-s − 5·32-s + 34-s − 12·37-s − 8·38-s + 12·41-s − 4·43-s + 4·44-s − 2·47-s + 5·50-s + 52-s + 10·53-s + 8·58-s + 10·59-s − 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.277·13-s − 1/4·16-s − 0.242·17-s + 1.83·19-s + 0.852·22-s − 25-s + 0.196·26-s − 1.48·29-s + 1.43·31-s − 0.883·32-s + 0.171·34-s − 1.97·37-s − 1.29·38-s + 1.87·41-s − 0.609·43-s + 0.603·44-s − 0.291·47-s + 0.707·50-s + 0.138·52-s + 1.37·53-s + 1.05·58-s + 1.30·59-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.87194306005156, −13.58189478030857, −13.18741527534835, −12.54944088747760, −12.10336464210944, −11.38839666902589, −11.10167352897526, −10.35926060946519, −9.952114875497428, −9.691834362864411, −9.094251354819219, −8.550018565341553, −8.015939638905197, −7.588865180396041, −7.259861206756946, −6.572862877958209, −5.619480057277940, −5.308007350095732, −4.972481870068780, −4.025418414938554, −3.711933380544892, −2.822737159433130, −2.257625059852536, −1.471702263228152, −0.7174832268398778, 0,
0.7174832268398778, 1.471702263228152, 2.257625059852536, 2.822737159433130, 3.711933380544892, 4.025418414938554, 4.972481870068780, 5.308007350095732, 5.619480057277940, 6.572862877958209, 7.259861206756946, 7.588865180396041, 8.015939638905197, 8.550018565341553, 9.094251354819219, 9.691834362864411, 9.952114875497428, 10.35926060946519, 11.10167352897526, 11.38839666902589, 12.10336464210944, 12.54944088747760, 13.18741527534835, 13.58189478030857, 13.87194306005156
