| L(s) = 1 | + 512·2-s + 2.46e4·3-s + 2.62e5·4-s − 1.95e6·5-s + 1.26e7·6-s − 1.71e8·7-s + 1.34e8·8-s − 5.55e8·9-s − 1.00e9·10-s − 1.18e10·11-s + 6.45e9·12-s + 3.59e10·13-s − 8.80e10·14-s − 4.81e10·15-s + 6.87e10·16-s − 7.46e11·17-s − 2.84e11·18-s + 2.68e12·19-s − 5.12e11·20-s − 4.23e12·21-s − 6.07e12·22-s + 1.59e10·23-s + 3.30e12·24-s + 3.81e12·25-s + 1.84e13·26-s − 4.23e13·27-s − 4.50e13·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.722·3-s + 1/2·4-s − 0.447·5-s + 0.511·6-s − 1.61·7-s + 0.353·8-s − 0.477·9-s − 0.316·10-s − 1.51·11-s + 0.361·12-s + 0.940·13-s − 1.13·14-s − 0.323·15-s + 1/4·16-s − 1.52·17-s − 0.337·18-s + 1.90·19-s − 0.223·20-s − 1.16·21-s − 1.07·22-s + 0.00184·23-s + 0.255·24-s + 1/5·25-s + 0.665·26-s − 1.06·27-s − 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{9} T \) |
| 5 | \( 1 + p^{9} T \) |
| good | 3 | \( 1 - 2738 p^{2} T + p^{19} T^{2} \) |
| 7 | \( 1 + 3508186 p^{2} T + p^{19} T^{2} \) |
| 11 | \( 1 + 11873833248 T + p^{19} T^{2} \) |
| 13 | \( 1 - 2767074434 p T + p^{19} T^{2} \) |
| 17 | \( 1 + 746262181734 T + p^{19} T^{2} \) |
| 19 | \( 1 - 2682185926820 T + p^{19} T^{2} \) |
| 23 | \( 1 - 15924407922 T + p^{19} T^{2} \) |
| 29 | \( 1 + 106190769937650 T + p^{19} T^{2} \) |
| 31 | \( 1 + 158223244950508 T + p^{19} T^{2} \) |
| 37 | \( 1 + 80246066291254 T + p^{19} T^{2} \) |
| 41 | \( 1 - 1895936900328342 T + p^{19} T^{2} \) |
| 43 | \( 1 - 3278663887209722 T + p^{19} T^{2} \) |
| 47 | \( 1 - 1296653846339646 T + p^{19} T^{2} \) |
| 53 | \( 1 - 12483704389153602 T + p^{19} T^{2} \) |
| 59 | \( 1 + 110746676451060300 T + p^{19} T^{2} \) |
| 61 | \( 1 + 17892488585662918 T + p^{19} T^{2} \) |
| 67 | \( 1 - 53295728410967486 T + p^{19} T^{2} \) |
| 71 | \( 1 - 406846488659414172 T + p^{19} T^{2} \) |
| 73 | \( 1 + 599276103913841518 T + p^{19} T^{2} \) |
| 79 | \( 1 + 183795029900265640 T + p^{19} T^{2} \) |
| 83 | \( 1 + 897306592327000998 T + p^{19} T^{2} \) |
| 89 | \( 1 - 498699120872108010 T + p^{19} T^{2} \) |
| 97 | \( 1 + 1088346985506567934 T + p^{19} T^{2} \) |
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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
−15.48777932411113350284835066331, −13.68007532873854245039035678766, −12.90358469162279056261009916109, −11.04594298499163138460326453502, −9.207525098723908525965157693690, −7.46465260744904713930753510960, −5.73933626269494695568558934148, −3.60532814218492723978423370442, −2.66449458175183241715677843249, 0,
2.66449458175183241715677843249, 3.60532814218492723978423370442, 5.73933626269494695568558934148, 7.46465260744904713930753510960, 9.207525098723908525965157693690, 11.04594298499163138460326453502, 12.90358469162279056261009916109, 13.68007532873854245039035678766, 15.48777932411113350284835066331
