| L(s) = 1 | − 3·3-s + 4·4-s + 5·5-s − 7-s + 9·9-s − 17·11-s − 12·12-s + 13·13-s − 15·15-s + 16·16-s + 31·17-s + 20·20-s + 3·21-s + 19·23-s + 25·25-s − 27·27-s − 4·28-s + 51·33-s − 5·35-s + 36·36-s − 61·37-s − 39·39-s + 43·41-s − 68·44-s + 45·45-s − 48·48-s − 48·49-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 5-s − 1/7·7-s + 9-s − 1.54·11-s − 12-s + 13-s − 15-s + 16-s + 1.82·17-s + 20-s + 1/7·21-s + 0.826·23-s + 25-s − 27-s − 1/7·28-s + 1.54·33-s − 1/7·35-s + 36-s − 1.64·37-s − 39-s + 1.04·41-s − 1.54·44-s + 45-s − 48-s − 0.979·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.639404000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.639404000\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + T + p^{2} T^{2} \) |
| 11 | \( 1 + 17 T + p^{2} T^{2} \) |
| 17 | \( 1 - 31 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( 1 - 19 T + p^{2} T^{2} \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 61 T + p^{2} T^{2} \) |
| 41 | \( 1 - 43 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 41 T + p^{2} T^{2} \) |
| 59 | \( 1 + 38 T + p^{2} T^{2} \) |
| 61 | \( 1 + 73 T + p^{2} T^{2} \) |
| 67 | \( 1 - 74 T + p^{2} T^{2} \) |
| 71 | \( 1 - 103 T + p^{2} T^{2} \) |
| 73 | \( 1 + 94 T + p^{2} T^{2} \) |
| 79 | \( 1 + 37 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 + 173 T + p^{2} T^{2} \) |
| 97 | \( 1 + 181 T + p^{2} T^{2} \) |
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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
−12.35650383321124473066500937930, −11.05414053390681389658879212792, −10.54074523906747723011572871732, −9.723815026965915891908854471809, −7.986562466275709437536075566327, −6.90572285915530245537702148843, −5.84822849403067246071761479192, −5.27727475927254964577085556976, −3.06763453592803290747245242836, −1.41025022659525055073707903900,
1.41025022659525055073707903900, 3.06763453592803290747245242836, 5.27727475927254964577085556976, 5.84822849403067246071761479192, 6.90572285915530245537702148843, 7.986562466275709437536075566327, 9.723815026965915891908854471809, 10.54074523906747723011572871732, 11.05414053390681389658879212792, 12.35650383321124473066500937930
