| L(s) = 1 | + 1.63e4·2-s − 4.33e5·3-s + 2.68e8·4-s − 7.10e9·6-s + 4.39e12·8-s − 2.26e13·9-s + 7.39e14·11-s − 1.16e14·12-s + 7.20e16·16-s − 2.28e17·17-s − 3.71e17·18-s + 1.56e18·19-s + 1.21e19·22-s − 1.90e18·24-s + 3.72e19·25-s + 1.97e19·27-s + 1.18e21·32-s − 3.20e20·33-s − 3.74e21·34-s − 6.09e21·36-s + 2.56e22·38-s + 3.52e22·41-s + 9.72e22·43-s + 1.98e23·44-s − 3.12e22·48-s + 4.59e23·49-s + 6.10e23·50-s + ⋯ |
| L(s) = 1 | + 2-s − 0.0906·3-s + 4-s − 0.0906·6-s + 8-s − 0.991·9-s + 1.94·11-s − 0.0906·12-s + 16-s − 1.35·17-s − 0.991·18-s + 1.96·19-s + 1.94·22-s − 0.0906·24-s + 25-s + 0.180·27-s + 32-s − 0.176·33-s − 1.35·34-s − 0.991·36-s + 1.96·38-s + 0.930·41-s + 1.31·43-s + 1.94·44-s − 0.0906·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+14) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{29}{2})\) |
\(\approx\) |
\(4.305672154\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.305672154\) |
| \(L(15)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{14} T \) |
| good | 3 | \( 1 + 433454 T + p^{28} T^{2} \) |
| 5 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 7 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 11 | \( 1 - 739138942594034 T + p^{28} T^{2} \) |
| 13 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 17 | \( 1 + 228753886708368574 T + p^{28} T^{2} \) |
| 19 | \( 1 - 1566734430868357394 T + p^{28} T^{2} \) |
| 23 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 29 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 31 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 37 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 41 | \( 1 - \)\(35\!\cdots\!34\)\( T + p^{28} T^{2} \) |
| 43 | \( 1 - \)\(97\!\cdots\!98\)\( T + p^{28} T^{2} \) |
| 47 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 53 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 59 | \( 1 - \)\(29\!\cdots\!22\)\( T + p^{28} T^{2} \) |
| 61 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 67 | \( 1 + \)\(66\!\cdots\!94\)\( T + p^{28} T^{2} \) |
| 71 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 73 | \( 1 + \)\(24\!\cdots\!14\)\( T + p^{28} T^{2} \) |
| 79 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 83 | \( 1 + \)\(50\!\cdots\!54\)\( T + p^{28} T^{2} \) |
| 89 | \( 1 + \)\(24\!\cdots\!26\)\( T + p^{28} T^{2} \) |
| 97 | \( 1 + \)\(91\!\cdots\!62\)\( T + p^{28} T^{2} \) |
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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
−14.67385592286726450153743673565, −13.77588237768469498216930707298, −12.04952496472542030999876608224, −11.17487668790532312416764181040, −9.063811212631746753491555361079, −7.02412706372284762160670867890, −5.78944673132899916059357021573, −4.23549690873169916812482701829, −2.86087718026792988946836221248, −1.17093338639934393638597003383,
1.17093338639934393638597003383, 2.86087718026792988946836221248, 4.23549690873169916812482701829, 5.78944673132899916059357021573, 7.02412706372284762160670867890, 9.063811212631746753491555361079, 11.17487668790532312416764181040, 12.04952496472542030999876608224, 13.77588237768469498216930707298, 14.67385592286726450153743673565
