| L(s) = 1 | + 48·3-s − 125·5-s − 1.64e3·7-s + 117·9-s − 172·11-s − 3.86e3·13-s − 6.00e3·15-s − 1.22e4·17-s + 2.59e4·19-s − 7.89e4·21-s + 1.29e4·23-s + 1.56e4·25-s − 9.93e4·27-s + 8.16e4·29-s − 1.56e5·31-s − 8.25e3·33-s + 2.05e5·35-s − 1.10e5·37-s − 1.85e5·39-s + 4.67e5·41-s + 4.99e5·43-s − 1.46e4·45-s − 3.96e5·47-s + 1.87e6·49-s − 5.88e5·51-s + 1.28e6·53-s + 2.15e4·55-s + ⋯ |
| L(s) = 1 | + 1.02·3-s − 0.447·5-s − 1.81·7-s + 0.0534·9-s − 0.0389·11-s − 0.487·13-s − 0.459·15-s − 0.604·17-s + 0.867·19-s − 1.85·21-s + 0.222·23-s + 1/5·25-s − 0.971·27-s + 0.621·29-s − 0.945·31-s − 0.0399·33-s + 0.810·35-s − 0.357·37-s − 0.500·39-s + 1.06·41-s + 0.957·43-s − 0.0239·45-s − 0.557·47-s + 2.28·49-s − 0.620·51-s + 1.18·53-s + 0.0174·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.559426520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559426520\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 - 16 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1644 T + p^{7} T^{2} \) |
| 11 | \( 1 + 172 T + p^{7} T^{2} \) |
| 13 | \( 1 + 3862 T + p^{7} T^{2} \) |
| 17 | \( 1 + 12254 T + p^{7} T^{2} \) |
| 19 | \( 1 - 25940 T + p^{7} T^{2} \) |
| 23 | \( 1 - 564 p T + p^{7} T^{2} \) |
| 29 | \( 1 - 81610 T + p^{7} T^{2} \) |
| 31 | \( 1 + 156888 T + p^{7} T^{2} \) |
| 37 | \( 1 + 110126 T + p^{7} T^{2} \) |
| 41 | \( 1 - 467882 T + p^{7} T^{2} \) |
| 43 | \( 1 - 499208 T + p^{7} T^{2} \) |
| 47 | \( 1 + 396884 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1280498 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1337420 T + p^{7} T^{2} \) |
| 61 | \( 1 - 923978 T + p^{7} T^{2} \) |
| 67 | \( 1 - 797304 T + p^{7} T^{2} \) |
| 71 | \( 1 - 5103392 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4267478 T + p^{7} T^{2} \) |
| 79 | \( 1 + 960 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6140832 T + p^{7} T^{2} \) |
| 89 | \( 1 - 2010570 T + p^{7} T^{2} \) |
| 97 | \( 1 + 4881934 T + p^{7} T^{2} \) |
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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
−10.13162299617811368029581234349, −9.371957971471402585297599000276, −8.725767779180783830068613400632, −7.54731977160391798175741050703, −6.78026660972686078230876078606, −5.58904095422811095091896412573, −4.00835488802654604122111920330, −3.17351561311018646626898923432, −2.42109502305476071984407007197, −0.54104954335733695571683375834,
0.54104954335733695571683375834, 2.42109502305476071984407007197, 3.17351561311018646626898923432, 4.00835488802654604122111920330, 5.58904095422811095091896412573, 6.78026660972686078230876078606, 7.54731977160391798175741050703, 8.725767779180783830068613400632, 9.371957971471402585297599000276, 10.13162299617811368029581234349
