| L(s) = 1 | − 2.15e24·3-s + 5.07e30·4-s − 2.15e43·7-s + 4.63e48·9-s − 1.09e55·12-s + 1.19e57·13-s + 2.57e61·16-s − 2.19e65·19-s + 4.64e67·21-s + 1.97e71·25-s − 9.98e72·27-s − 1.09e74·28-s − 2.23e76·31-s + 2.35e79·36-s + 9.77e79·37-s − 2.57e81·39-s + 2.37e83·43-s − 5.53e85·48-s + 3.06e86·49-s + 6.06e87·52-s + 4.73e89·57-s − 2.18e91·61-s − 9.99e91·63-s + 1.30e92·64-s − 2.61e93·67-s − 1.27e95·73-s − 4.24e95·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 1.71·7-s + 9-s − 12-s + 1.84·13-s + 16-s − 1.33·19-s + 1.71·21-s + 25-s − 27-s − 1.71·28-s − 1.95·31-s + 36-s + 1.02·37-s − 1.84·39-s + 1.17·43-s − 48-s + 1.93·49-s + 1.84·52-s + 1.33·57-s − 1.93·61-s − 1.71·63-s + 64-s − 1.94·67-s − 1.19·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(103-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+51) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{103}{2})\) |
\(\approx\) |
\(1.533036436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533036436\) |
| \(L(52)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{51} T \) |
| good | 2 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 5 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 7 | \( 1 + \)\(21\!\cdots\!86\)\( T + p^{102} T^{2} \) |
| 11 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 13 | \( 1 - \)\(11\!\cdots\!26\)\( T + p^{102} T^{2} \) |
| 17 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 19 | \( 1 + \)\(21\!\cdots\!62\)\( T + p^{102} T^{2} \) |
| 23 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 29 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 31 | \( 1 + \)\(22\!\cdots\!38\)\( T + p^{102} T^{2} \) |
| 37 | \( 1 - \)\(97\!\cdots\!74\)\( T + p^{102} T^{2} \) |
| 41 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 43 | \( 1 - \)\(23\!\cdots\!86\)\( T + p^{102} T^{2} \) |
| 47 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 53 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 59 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 61 | \( 1 + \)\(21\!\cdots\!78\)\( T + p^{102} T^{2} \) |
| 67 | \( 1 + \)\(26\!\cdots\!66\)\( T + p^{102} T^{2} \) |
| 71 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 73 | \( 1 + \)\(12\!\cdots\!54\)\( T + p^{102} T^{2} \) |
| 79 | \( 1 - \)\(48\!\cdots\!58\)\( T + p^{102} T^{2} \) |
| 83 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 89 | \( ( 1 - p^{51} T )( 1 + p^{51} T ) \) |
| 97 | \( 1 + \)\(21\!\cdots\!06\)\( T + p^{102} T^{2} \) |
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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
−11.05469864093145635847538760699, −10.49338188552047432878316604785, −9.055418626850451565840399587423, −7.25718135064059216682650577775, −6.24549916056331250729888853021, −5.95562942040864903670799669298, −4.03277002315548619289214768199, −3.04814721284226039869899058626, −1.63521002719601114357592607540, −0.55099578280587652703273091221,
0.55099578280587652703273091221, 1.63521002719601114357592607540, 3.04814721284226039869899058626, 4.03277002315548619289214768199, 5.95562942040864903670799669298, 6.24549916056331250729888853021, 7.25718135064059216682650577775, 9.055418626850451565840399587423, 10.49338188552047432878316604785, 11.05469864093145635847538760699
