| L(s) = 1 | − 5.04e4·2-s + 1.97e7·3-s + 1.46e9·4-s − 9.98e11·6-s − 1.98e13·8-s + 1.86e14·9-s + 2.90e16·12-s + 9.69e16·13-s − 5.75e17·16-s − 9.38e18·18-s − 2.66e20·23-s − 3.92e20·24-s + 9.31e20·25-s − 4.88e21·26-s − 3.91e20·27-s − 4.57e21·29-s + 3.15e22·31-s + 5.03e22·32-s + 2.73e23·36-s + 1.91e24·39-s + 3.10e24·41-s + 1.34e25·46-s − 1.77e25·47-s − 1.14e25·48-s + 2.25e25·49-s − 4.69e25·50-s + 1.42e26·52-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.37·3-s + 1.36·4-s − 2.12·6-s − 0.563·8-s + 0.904·9-s + 1.88·12-s + 1.89·13-s − 0.499·16-s − 1.39·18-s − 23-s − 0.777·24-s + 25-s − 2.91·26-s − 0.132·27-s − 0.530·29-s + 1.34·31-s + 1.33·32-s + 1.23·36-s + 2.61·39-s + 1.99·41-s + 1.53·46-s − 1.46·47-s − 0.689·48-s + 49-s − 1.53·50-s + 2.58·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{31}{2})\) |
\(\approx\) |
\(1.981336616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981336616\) |
| \(L(16)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + p^{15} T \) |
| good | 2 | \( 1 + 50407 T + p^{30} T^{2} \) |
| 3 | \( 1 - 19799482 T + p^{30} T^{2} \) |
| 5 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 7 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 11 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 13 | \( 1 - 96954465195037082 T + p^{30} T^{2} \) |
| 17 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 19 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 29 | \( 1 + \)\(45\!\cdots\!74\)\( T + p^{30} T^{2} \) |
| 31 | \( 1 - \)\(31\!\cdots\!74\)\( T + p^{30} T^{2} \) |
| 37 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 41 | \( 1 - \)\(31\!\cdots\!74\)\( T + p^{30} T^{2} \) |
| 43 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 47 | \( 1 + \)\(17\!\cdots\!82\)\( T + p^{30} T^{2} \) |
| 53 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 59 | \( 1 - \)\(13\!\cdots\!98\)\( T + p^{30} T^{2} \) |
| 61 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 67 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 71 | \( 1 + \)\(32\!\cdots\!26\)\( T + p^{30} T^{2} \) |
| 73 | \( 1 + \)\(50\!\cdots\!18\)\( T + p^{30} T^{2} \) |
| 79 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 83 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 89 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
| 97 | \( ( 1 - p^{15} T )( 1 + p^{15} T ) \) |
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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
−11.32179138845266023916613259846, −10.20636451449380484021649492062, −9.081666179065423840549007780265, −8.464331808217308924425201463959, −7.65872226315270299905087521141, −6.27255152329906035879746329481, −4.04866669702370114220443149468, −2.82950479287640721629202862451, −1.72565852388529464991066952301, −0.795779337645979707375269002458,
0.795779337645979707375269002458, 1.72565852388529464991066952301, 2.82950479287640721629202862451, 4.04866669702370114220443149468, 6.27255152329906035879746329481, 7.65872226315270299905087521141, 8.464331808217308924425201463959, 9.081666179065423840549007780265, 10.20636451449380484021649492062, 11.32179138845266023916613259846
