| L(s) = 1 | + 8·2-s − 72.4·3-s + 64·4-s + 226.·5-s − 579.·6-s + 1.75e3·7-s + 512·8-s + 3.06e3·9-s + 1.81e3·10-s + 1.33e3·11-s − 4.63e3·12-s − 9.90e3·13-s + 1.40e4·14-s − 1.64e4·15-s + 4.09e3·16-s + 2.19e4·17-s + 2.45e4·18-s + 3.79e4·19-s + 1.44e4·20-s − 1.26e5·21-s + 1.06e4·22-s + 8.90e3·23-s − 3.71e4·24-s − 2.68e4·25-s − 7.92e4·26-s − 6.38e4·27-s + 1.12e5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.55·3-s + 0.5·4-s + 0.810·5-s − 1.09·6-s + 1.92·7-s + 0.353·8-s + 1.40·9-s + 0.572·10-s + 0.301·11-s − 0.775·12-s − 1.24·13-s + 1.36·14-s − 1.25·15-s + 0.250·16-s + 1.08·17-s + 0.992·18-s + 1.27·19-s + 0.405·20-s − 2.99·21-s + 0.213·22-s + 0.152·23-s − 0.548·24-s − 0.343·25-s − 0.883·26-s − 0.624·27-s + 0.964·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.026409668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026409668\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 11 | \( 1 - 1.33e3T \) |
| good | 3 | \( 1 + 72.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 226.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.75e3T + 8.23e5T^{2} \) |
| 13 | \( 1 + 9.90e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.19e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.79e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 8.90e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.48e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 6.94e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.07e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.26e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.53e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.26e4T + 1.17e12T^{2} \) |
| 59 | \( 1 + 4.28e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.03e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 6.98e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.00e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.84e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.47e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.55e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.39e7T + 8.07e13T^{2} \) |
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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
−16.78140729005711663746400424471, −14.95222047498426488598763333328, −13.88789065474692011749096096171, −12.09253308708077720115420804368, −11.47286221488653987237158568944, −10.11131630818385781309617493968, −7.43542258591045502889278444378, −5.59929163749389598532632775262, −4.91297575058219524071789527774, −1.51038172733195659926201817794,
1.51038172733195659926201817794, 4.91297575058219524071789527774, 5.59929163749389598532632775262, 7.43542258591045502889278444378, 10.11131630818385781309617493968, 11.47286221488653987237158568944, 12.09253308708077720115420804368, 13.88789065474692011749096096171, 14.95222047498426488598763333328, 16.78140729005711663746400424471
