| L(s) = 1 | − 6.08e35·3-s + 1.42e45·4-s + 4.68e63·7-s + 3.69e71·9-s − 8.68e80·12-s + 1.78e83·13-s + 2.03e90·16-s + 5.58e94·19-s − 2.84e99·21-s + 7.00e104·25-s − 2.25e107·27-s + 6.67e108·28-s − 1.92e111·31-s + 5.28e116·36-s − 8.04e117·37-s − 1.08e119·39-s + 3.39e122·43-s − 1.23e126·48-s + 1.60e127·49-s + 2.54e128·52-s − 3.39e130·57-s + 1.56e134·61-s + 1.73e135·63-s + 2.90e135·64-s + 1.52e137·67-s − 9.96e139·73-s − 4.26e140·75-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s + 1.94·7-s + 9-s − 12-s + 0.506·13-s + 16-s + 0.0692·19-s − 1.94·21-s + 25-s − 27-s + 1.94·28-s − 0.270·31-s + 36-s − 1.95·37-s − 0.506·39-s + 1.04·43-s − 48-s + 2.76·49-s + 0.506·52-s − 0.0692·57-s + 1.96·61-s + 1.94·63-s + 64-s + 1.68·67-s − 1.77·73-s − 75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(151-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+75) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{151}{2})\) |
\(\approx\) |
\(3.832638298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.832638298\) |
| \(L(76)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{75} T \) |
| good | 2 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 5 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 7 | \( 1 - \)\(46\!\cdots\!14\)\( T + p^{150} T^{2} \) |
| 11 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 13 | \( 1 - \)\(17\!\cdots\!86\)\( T + p^{150} T^{2} \) |
| 17 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 19 | \( 1 - \)\(55\!\cdots\!98\)\( T + p^{150} T^{2} \) |
| 23 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 29 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 31 | \( 1 + \)\(19\!\cdots\!98\)\( T + p^{150} T^{2} \) |
| 37 | \( 1 + \)\(80\!\cdots\!86\)\( T + p^{150} T^{2} \) |
| 41 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 43 | \( 1 - \)\(33\!\cdots\!86\)\( T + p^{150} T^{2} \) |
| 47 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 53 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 59 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 61 | \( 1 - \)\(15\!\cdots\!02\)\( T + p^{150} T^{2} \) |
| 67 | \( 1 - \)\(15\!\cdots\!14\)\( T + p^{150} T^{2} \) |
| 71 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 73 | \( 1 + \)\(99\!\cdots\!14\)\( T + p^{150} T^{2} \) |
| 79 | \( 1 - \)\(36\!\cdots\!98\)\( T + p^{150} T^{2} \) |
| 83 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 89 | \( ( 1 - p^{75} T )( 1 + p^{75} T ) \) |
| 97 | \( 1 + \)\(14\!\cdots\!86\)\( T + p^{150} T^{2} \) |
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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
−10.37139468410155454018159278000, −8.598465383801243001437808401850, −7.56644893614498954947337680691, −6.75800565603168479015135068310, −5.58639135917123429596291484107, −4.94755810409614344978303226131, −3.79180387577606116046795468133, −2.24758975344058497078215039020, −1.47972729184382218881690380597, −0.814635545003965900100935715849,
0.814635545003965900100935715849, 1.47972729184382218881690380597, 2.24758975344058497078215039020, 3.79180387577606116046795468133, 4.94755810409614344978303226131, 5.58639135917123429596291484107, 6.75800565603168479015135068310, 7.56644893614498954947337680691, 8.598465383801243001437808401850, 10.37139468410155454018159278000
