| L(s) = 1 | − 1.31e5·2-s + 1.29e8·3-s + 1.71e10·4-s − 4.23e11·5-s − 1.69e13·6-s − 3.80e14·7-s − 2.25e15·8-s + 1.66e16·9-s + 5.55e16·10-s − 9.96e17·11-s + 2.21e18·12-s + 4.99e19·14-s − 5.47e19·15-s + 2.95e20·16-s − 2.18e21·18-s − 7.28e21·20-s − 4.91e22·21-s + 1.30e23·22-s − 2.90e23·24-s − 4.02e23·25-s + 2.15e24·27-s − 6.54e24·28-s − 1.34e25·29-s + 7.17e24·30-s − 4.40e25·31-s − 3.86e25·32-s − 1.28e26·33-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 0.555·5-s − 6-s − 1.63·7-s − 8-s + 9-s + 0.555·10-s − 1.97·11-s + 12-s + 1.63·14-s − 0.555·15-s + 16-s − 18-s − 0.555·20-s − 1.63·21-s + 1.97·22-s − 24-s − 0.691·25-s + 27-s − 1.63·28-s − 1.84·29-s + 0.555·30-s − 1.95·31-s − 32-s − 1.97·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{35}{2})\) |
\(\approx\) |
\(0.2295841752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2295841752\) |
| \(L(18)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{17} T \) |
| 3 | \( 1 - p^{17} T \) |
| good | 5 | \( 1 + 423872734078 T + p^{34} T^{2} \) |
| 7 | \( 1 + 380838707036170 T + p^{34} T^{2} \) |
| 11 | \( 1 + 996717635485503370 T + p^{34} T^{2} \) |
| 13 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 17 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 19 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 23 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 29 | \( 1 + \)\(13\!\cdots\!50\)\( T + p^{34} T^{2} \) |
| 31 | \( 1 + \)\(44\!\cdots\!22\)\( T + p^{34} T^{2} \) |
| 37 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 41 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 43 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 47 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 53 | \( 1 - \)\(12\!\cdots\!26\)\( T + p^{34} T^{2} \) |
| 59 | \( 1 + \)\(25\!\cdots\!70\)\( T + p^{34} T^{2} \) |
| 61 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 67 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 71 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 73 | \( 1 + \)\(31\!\cdots\!50\)\( T + p^{34} T^{2} \) |
| 79 | \( 1 + \)\(38\!\cdots\!18\)\( T + p^{34} T^{2} \) |
| 83 | \( 1 + \)\(21\!\cdots\!46\)\( T + p^{34} T^{2} \) |
| 89 | \( ( 1 - p^{17} T )( 1 + p^{17} T ) \) |
| 97 | \( 1 - \)\(11\!\cdots\!70\)\( T + p^{34} T^{2} \) |
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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
−10.82423406614799591530532343325, −9.866247853052822918024119470993, −9.040208411626343383712878203839, −7.77951105358394591533533197246, −7.21938378394378036128354997590, −5.72153565189302160076728093725, −3.66047678436437355723427819001, −2.91299316254967153407391410149, −1.95074914824107865018542942152, −0.20993534257199541537031882562,
0.20993534257199541537031882562, 1.95074914824107865018542942152, 2.91299316254967153407391410149, 3.66047678436437355723427819001, 5.72153565189302160076728093725, 7.21938378394378036128354997590, 7.77951105358394591533533197246, 9.040208411626343383712878203839, 9.866247853052822918024119470993, 10.82423406614799591530532343325
