| L(s) = 1 | − 5·3-s + 64·4-s + 191·5-s + 155·7-s − 704·9-s − 320·12-s − 955·15-s + 4.09e3·16-s + 6.05e3·17-s + 443·19-s + 1.22e4·20-s − 775·21-s + 2.08e4·25-s + 7.16e3·27-s + 9.92e3·28-s − 2.34e4·29-s + 2.96e4·35-s − 4.50e4·36-s + 1.37e5·41-s − 1.34e5·45-s − 2.04e4·48-s − 9.36e4·49-s − 3.02e4·51-s − 1.90e5·53-s − 2.21e3·57-s − 2.05e5·59-s − 6.11e4·60-s + ⋯ |
| L(s) = 1 | − 0.185·3-s + 4-s + 1.52·5-s + 0.451·7-s − 0.965·9-s − 0.185·12-s − 0.282·15-s + 16-s + 1.23·17-s + 0.0645·19-s + 1.52·20-s − 0.0836·21-s + 1.33·25-s + 0.364·27-s + 0.451·28-s − 0.963·29-s + 0.690·35-s − 0.965·36-s + 1.99·41-s − 1.47·45-s − 0.185·48-s − 0.795·49-s − 0.228·51-s − 1.28·53-s − 0.0119·57-s − 59-s − 0.282·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.737033552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.737033552\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 3 | \( 1 + 5 T + p^{6} T^{2} \) |
| 5 | \( 1 - 191 T + p^{6} T^{2} \) |
| 7 | \( 1 - 155 T + p^{6} T^{2} \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( 1 - 6050 T + p^{6} T^{2} \) |
| 19 | \( 1 - 443 T + p^{6} T^{2} \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 + 23497 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 41 | \( 1 - 137783 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( 1 + 190825 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( 1 + 683422 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( 1 + 288853 T + p^{6} T^{2} \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 97 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
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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
−14.10584551369521964542591014881, −12.70824428199690825193982961526, −11.48717184455605153560599841218, −10.52034171822121839564032932065, −9.355985583408853208116101962119, −7.75698367492750124051403542193, −6.18499513177944521331361087885, −5.46802252947789690290386787815, −2.82786542540259953009667312199, −1.51537091767807603673283425594,
1.51537091767807603673283425594, 2.82786542540259953009667312199, 5.46802252947789690290386787815, 6.18499513177944521331361087885, 7.75698367492750124051403542193, 9.355985583408853208116101962119, 10.52034171822121839564032932065, 11.48717184455605153560599841218, 12.70824428199690825193982961526, 14.10584551369521964542591014881
