| L(s) = 1 | − 8·2-s − 39·3-s + 64·4-s + 385·5-s + 312·6-s − 293·7-s − 512·8-s − 666·9-s − 3.08e3·10-s − 5.40e3·11-s − 2.49e3·12-s + 2.19e3·13-s + 2.34e3·14-s − 1.50e4·15-s + 4.09e3·16-s − 2.10e4·17-s + 5.32e3·18-s − 2.73e4·19-s + 2.46e4·20-s + 1.14e4·21-s + 4.32e4·22-s − 6.30e4·23-s + 1.99e4·24-s + 7.01e4·25-s − 1.75e4·26-s + 1.11e5·27-s − 1.87e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.833·3-s + 1/2·4-s + 1.37·5-s + 0.589·6-s − 0.322·7-s − 0.353·8-s − 0.304·9-s − 0.973·10-s − 1.22·11-s − 0.416·12-s + 0.277·13-s + 0.228·14-s − 1.14·15-s + 1/4·16-s − 1.03·17-s + 0.215·18-s − 0.913·19-s + 0.688·20-s + 0.269·21-s + 0.865·22-s − 1.08·23-s + 0.294·24-s + 0.897·25-s − 0.196·26-s + 1.08·27-s − 0.161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{3} T \) |
| 13 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 + 13 p T + p^{7} T^{2} \) |
| 5 | \( 1 - 77 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 293 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5402 T + p^{7} T^{2} \) |
| 17 | \( 1 + 21011 T + p^{7} T^{2} \) |
| 19 | \( 1 + 27326 T + p^{7} T^{2} \) |
| 23 | \( 1 + 63072 T + p^{7} T^{2} \) |
| 29 | \( 1 - 122238 T + p^{7} T^{2} \) |
| 31 | \( 1 + 208396 T + p^{7} T^{2} \) |
| 37 | \( 1 + 442379 T + p^{7} T^{2} \) |
| 41 | \( 1 - 58000 T + p^{7} T^{2} \) |
| 43 | \( 1 + 202025 T + p^{7} T^{2} \) |
| 47 | \( 1 - 588511 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1684336 T + p^{7} T^{2} \) |
| 59 | \( 1 + 442630 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1083608 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3443486 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2084705 T + p^{7} T^{2} \) |
| 73 | \( 1 - 5937890 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6609256 T + p^{7} T^{2} \) |
| 83 | \( 1 + 142740 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6985286 T + p^{7} T^{2} \) |
| 97 | \( 1 + 200762 T + p^{7} T^{2} \) |
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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
−15.66885780145088908854242561630, −13.86971349348151742307210578910, −12.62284679773150127884684477029, −10.94369025978432517939191168616, −10.10492723676852036712860160298, −8.626992702002756715150582372352, −6.54518408814233103989716025511, −5.47685771276203998293924043969, −2.22677345895488773830110414999, 0,
2.22677345895488773830110414999, 5.47685771276203998293924043969, 6.54518408814233103989716025511, 8.626992702002756715150582372352, 10.10492723676852036712860160298, 10.94369025978432517939191168616, 12.62284679773150127884684477029, 13.86971349348151742307210578910, 15.66885780145088908854242561630
