| L(s) = 1 | − 78·2-s + 243·3-s + 4.03e3·4-s − 1.89e4·6-s + 2.77e4·7-s − 1.55e5·8-s + 5.90e4·9-s + 6.37e5·11-s + 9.80e5·12-s − 7.66e5·13-s − 2.16e6·14-s + 3.82e6·16-s − 3.08e6·17-s − 4.60e6·18-s − 1.95e7·19-s + 6.74e6·21-s − 4.97e7·22-s − 1.53e7·23-s − 3.76e7·24-s + 5.97e7·26-s + 1.43e7·27-s + 1.12e8·28-s + 1.07e7·29-s − 5.09e7·31-s + 1.88e7·32-s + 1.54e8·33-s + 2.40e8·34-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 0.577·3-s + 1.97·4-s − 0.995·6-s + 0.624·7-s − 1.67·8-s + 1/3·9-s + 1.19·11-s + 1.13·12-s − 0.572·13-s − 1.07·14-s + 0.912·16-s − 0.526·17-s − 0.574·18-s − 1.80·19-s + 0.360·21-s − 2.05·22-s − 0.496·23-s − 0.965·24-s + 0.986·26-s + 0.192·27-s + 1.23·28-s + 0.0973·29-s − 0.319·31-s + 0.0995·32-s + 0.689·33-s + 0.908·34-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 | \( 1 + 39 p T + p^{11} T^{2} \) |
| 7 | \( 1 - 27760 T + p^{11} T^{2} \) |
| 11 | \( 1 - 637836 T + p^{11} T^{2} \) |
| 13 | \( 1 + 766214 T + p^{11} T^{2} \) |
| 17 | \( 1 + 3084354 T + p^{11} T^{2} \) |
| 19 | \( 1 + 1026916 p T + p^{11} T^{2} \) |
| 23 | \( 1 + 15312360 T + p^{11} T^{2} \) |
| 29 | \( 1 - 10751262 T + p^{11} T^{2} \) |
| 31 | \( 1 + 50937400 T + p^{11} T^{2} \) |
| 37 | \( 1 + 664740830 T + p^{11} T^{2} \) |
| 41 | \( 1 - 898833450 T + p^{11} T^{2} \) |
| 43 | \( 1 - 957947188 T + p^{11} T^{2} \) |
| 47 | \( 1 - 1555741344 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3792417030 T + p^{11} T^{2} \) |
| 59 | \( 1 - 555306924 T + p^{11} T^{2} \) |
| 61 | \( 1 - 4950420998 T + p^{11} T^{2} \) |
| 67 | \( 1 + 5292399284 T + p^{11} T^{2} \) |
| 71 | \( 1 + 14831086248 T + p^{11} T^{2} \) |
| 73 | \( 1 + 13971005210 T + p^{11} T^{2} \) |
| 79 | \( 1 - 3720542360 T + p^{11} T^{2} \) |
| 83 | \( 1 + 8768454036 T + p^{11} T^{2} \) |
| 89 | \( 1 + 25472769174 T + p^{11} T^{2} \) |
| 97 | \( 1 - 39092494846 T + p^{11} T^{2} \) |
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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
−11.28145768030827289012038987002, −10.34964114551710110764799885071, −9.175358424293325375190342143504, −8.564744993980433753542775631165, −7.47997417903461541209968917362, −6.43006448255509703298124724928, −4.23574145463521176079413800509, −2.33161597608196287437532485479, −1.45251746781522323562384346678, 0,
1.45251746781522323562384346678, 2.33161597608196287437532485479, 4.23574145463521176079413800509, 6.43006448255509703298124724928, 7.47997417903461541209968917362, 8.564744993980433753542775631165, 9.175358424293325375190342143504, 10.34964114551710110764799885071, 11.28145768030827289012038987002
