| L(s) = 1 | + 56·2-s + 243·3-s + 1.08e3·4-s + 1.36e4·6-s − 2.79e4·7-s − 5.37e4·8-s + 5.90e4·9-s − 1.12e5·11-s + 2.64e5·12-s + 1.09e6·13-s − 1.56e6·14-s − 5.23e6·16-s + 2.49e5·17-s + 3.30e6·18-s − 1.37e7·19-s − 6.80e6·21-s − 6.27e6·22-s − 4.13e7·23-s − 1.30e7·24-s + 6.14e7·26-s + 1.43e7·27-s − 3.04e7·28-s − 4.53e6·29-s − 2.65e8·31-s − 1.83e8·32-s − 2.72e7·33-s + 1.39e7·34-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 0.577·3-s + 0.531·4-s + 0.714·6-s − 0.629·7-s − 0.580·8-s + 1/3·9-s − 0.209·11-s + 0.306·12-s + 0.819·13-s − 0.778·14-s − 1.24·16-s + 0.0426·17-s + 0.412·18-s − 1.27·19-s − 0.363·21-s − 0.259·22-s − 1.34·23-s − 0.334·24-s + 1.01·26-s + 0.192·27-s − 0.334·28-s − 0.0410·29-s − 1.66·31-s − 0.965·32-s − 0.121·33-s + 0.0527·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{5} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 7 p^{3} T + p^{11} T^{2} \) |
| 7 | \( 1 + 27984 T + p^{11} T^{2} \) |
| 11 | \( 1 + 112028 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1096922 T + p^{11} T^{2} \) |
| 17 | \( 1 - 249566 T + p^{11} T^{2} \) |
| 19 | \( 1 + 13712420 T + p^{11} T^{2} \) |
| 23 | \( 1 + 41395728 T + p^{11} T^{2} \) |
| 29 | \( 1 + 4533850 T + p^{11} T^{2} \) |
| 31 | \( 1 + 265339008 T + p^{11} T^{2} \) |
| 37 | \( 1 - 212136946 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1266969958 T + p^{11} T^{2} \) |
| 43 | \( 1 + 14129548 T + p^{11} T^{2} \) |
| 47 | \( 1 - 2657273336 T + p^{11} T^{2} \) |
| 53 | \( 1 + 2402699278 T + p^{11} T^{2} \) |
| 59 | \( 1 - 7498737220 T + p^{11} T^{2} \) |
| 61 | \( 1 + 4064828858 T + p^{11} T^{2} \) |
| 67 | \( 1 + 6871514244 T + p^{11} T^{2} \) |
| 71 | \( 1 + 13283734648 T + p^{11} T^{2} \) |
| 73 | \( 1 - 28875844262 T + p^{11} T^{2} \) |
| 79 | \( 1 - 27100302240 T + p^{11} T^{2} \) |
| 83 | \( 1 - 34365255132 T + p^{11} T^{2} \) |
| 89 | \( 1 + 63500412630 T + p^{11} T^{2} \) |
| 97 | \( 1 + 19634495234 T + p^{11} T^{2} \) |
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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
−12.17242815634488932142402820516, −10.75785838189625702982584409499, −9.399438827494477269482114777896, −8.300528101871064586823089217194, −6.68711566786463995661643809590, −5.67680545633352270658412311293, −4.19441758888477393522441742971, −3.38698039418375951758856280625, −2.07780812821752574083339430050, 0,
2.07780812821752574083339430050, 3.38698039418375951758856280625, 4.19441758888477393522441742971, 5.67680545633352270658412311293, 6.68711566786463995661643809590, 8.300528101871064586823089217194, 9.399438827494477269482114777896, 10.75785838189625702982584409499, 12.17242815634488932142402820516
