| L(s) = 1 | + 8·2-s + 28·3-s + 64·4-s + 125·5-s + 224·6-s + 104·7-s + 512·8-s − 1.40e3·9-s + 1.00e3·10-s − 5.14e3·11-s + 1.79e3·12-s − 8.60e3·13-s + 832·14-s + 3.50e3·15-s + 4.09e3·16-s + 2.02e4·17-s − 1.12e4·18-s + 4.55e4·19-s + 8.00e3·20-s + 2.91e3·21-s − 4.11e4·22-s − 7.20e4·23-s + 1.43e4·24-s + 1.56e4·25-s − 6.88e4·26-s − 1.00e5·27-s + 6.65e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.598·3-s + 1/2·4-s + 0.447·5-s + 0.423·6-s + 0.114·7-s + 0.353·8-s − 0.641·9-s + 0.316·10-s − 1.16·11-s + 0.299·12-s − 1.08·13-s + 0.0810·14-s + 0.267·15-s + 1/4·16-s + 1.00·17-s − 0.453·18-s + 1.52·19-s + 0.223·20-s + 0.0686·21-s − 0.824·22-s − 1.23·23-s + 0.211·24-s + 1/5·25-s − 0.767·26-s − 0.982·27-s + 0.0573·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.291724059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.291724059\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| good | 3 | \( 1 - 28 T + p^{7} T^{2} \) |
| 7 | \( 1 - 104 T + p^{7} T^{2} \) |
| 11 | \( 1 + 468 p T + p^{7} T^{2} \) |
| 13 | \( 1 + 8602 T + p^{7} T^{2} \) |
| 17 | \( 1 - 20274 T + p^{7} T^{2} \) |
| 19 | \( 1 - 45500 T + p^{7} T^{2} \) |
| 23 | \( 1 + 72072 T + p^{7} T^{2} \) |
| 29 | \( 1 - 231510 T + p^{7} T^{2} \) |
| 31 | \( 1 + 80128 T + p^{7} T^{2} \) |
| 37 | \( 1 - 104654 T + p^{7} T^{2} \) |
| 41 | \( 1 - 584922 T + p^{7} T^{2} \) |
| 43 | \( 1 + 795532 T + p^{7} T^{2} \) |
| 47 | \( 1 - 425664 T + p^{7} T^{2} \) |
| 53 | \( 1 - 1500798 T + p^{7} T^{2} \) |
| 59 | \( 1 - 246420 T + p^{7} T^{2} \) |
| 61 | \( 1 - 893942 T + p^{7} T^{2} \) |
| 67 | \( 1 + 2336836 T + p^{7} T^{2} \) |
| 71 | \( 1 + 203688 T + p^{7} T^{2} \) |
| 73 | \( 1 + 3805702 T + p^{7} T^{2} \) |
| 79 | \( 1 - 5053040 T + p^{7} T^{2} \) |
| 83 | \( 1 + 45492 T + p^{7} T^{2} \) |
| 89 | \( 1 - 980010 T + p^{7} T^{2} \) |
| 97 | \( 1 + 5247646 T + p^{7} T^{2} \) |
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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
−19.69803505129928307001257753087, −17.94412345750497628085197694037, −16.22508016033046783568158432420, −14.62868265588610706563067552433, −13.67682192464925168364077324781, −12.03419118100529093390830307267, −10.00092841234296038945738009967, −7.81432905023217102108099172222, −5.40502895146420994065255763874, −2.76264866391344794513661279737,
2.76264866391344794513661279737, 5.40502895146420994065255763874, 7.81432905023217102108099172222, 10.00092841234296038945738009967, 12.03419118100529093390830307267, 13.67682192464925168364077324781, 14.62868265588610706563067552433, 16.22508016033046783568158432420, 17.94412345750497628085197694037, 19.69803505129928307001257753087
