| L(s) = 1 | − 1.63e4·2-s − 4.78e6·3-s + 2.68e8·4-s − 1.15e10·5-s + 7.83e10·6-s − 1.93e11·7-s − 4.39e12·8-s + 2.28e13·9-s + 1.89e14·10-s + 7.20e14·11-s − 1.28e15·12-s + 3.16e15·14-s + 5.53e16·15-s + 7.20e16·16-s − 3.74e17·18-s − 3.10e18·20-s + 9.23e17·21-s − 1.18e19·22-s + 2.10e19·24-s + 9.67e19·25-s − 1.09e20·27-s − 5.18e19·28-s − 2.39e20·29-s − 9.07e20·30-s + 1.48e21·31-s − 1.18e21·32-s − 3.44e21·33-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 4-s − 1.89·5-s + 6-s − 0.284·7-s − 8-s + 9-s + 1.89·10-s + 1.89·11-s − 12-s + 0.284·14-s + 1.89·15-s + 16-s − 18-s − 1.89·20-s + 0.284·21-s − 1.89·22-s + 24-s + 2.59·25-s − 27-s − 0.284·28-s − 0.803·29-s − 1.89·30-s + 1.96·31-s − 32-s − 1.89·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+14) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{29}{2})\) |
\(\approx\) |
\(0.5235894953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5235894953\) |
| \(L(15)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{14} T \) |
| 3 | \( 1 + p^{14} T \) |
| good | 5 | \( 1 + 11577394034 T + p^{28} T^{2} \) |
| 7 | \( 1 + 193132900798 T + p^{28} T^{2} \) |
| 11 | \( 1 - 720252825193582 T + p^{28} T^{2} \) |
| 13 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 17 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 19 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 23 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 29 | \( 1 + \)\(23\!\cdots\!38\)\( T + p^{28} T^{2} \) |
| 31 | \( 1 - \)\(14\!\cdots\!42\)\( T + p^{28} T^{2} \) |
| 37 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 41 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 43 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 47 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 53 | \( 1 + \)\(24\!\cdots\!38\)\( T + p^{28} T^{2} \) |
| 59 | \( 1 - \)\(46\!\cdots\!22\)\( T + p^{28} T^{2} \) |
| 61 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 67 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 71 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 73 | \( 1 + \)\(43\!\cdots\!18\)\( T + p^{28} T^{2} \) |
| 79 | \( 1 + \)\(38\!\cdots\!38\)\( T + p^{28} T^{2} \) |
| 83 | \( 1 - \)\(12\!\cdots\!42\)\( T + p^{28} T^{2} \) |
| 89 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 97 | \( 1 + \)\(12\!\cdots\!38\)\( T + p^{28} T^{2} \) |
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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
−11.71479730201141467930988056358, −11.05232284700235042626195579740, −9.581851619807778250020850276609, −8.274750305317749161032169296805, −7.11916532416645354058509089188, −6.35516339218918049637987360888, −4.41139875107295382322538765261, −3.40193710221030931823047710389, −1.31295400157381671026796829343, −0.47115282267813737411299623663,
0.47115282267813737411299623663, 1.31295400157381671026796829343, 3.40193710221030931823047710389, 4.41139875107295382322538765261, 6.35516339218918049637987360888, 7.11916532416645354058509089188, 8.274750305317749161032169296805, 9.581851619807778250020850276609, 11.05232284700235042626195579740, 11.71479730201141467930988056358
