L(s) = 1 | + 16·2-s + 137·3-s + 256·4-s − 595·5-s + 2.19e3·6-s + 1.13e4·7-s + 4.09e3·8-s − 914·9-s − 9.52e3·10-s − 1.46e4·11-s + 3.50e4·12-s + 5.56e4·13-s + 1.81e5·14-s − 8.15e4·15-s + 6.55e4·16-s + 4.21e5·17-s − 1.46e4·18-s − 4.35e5·19-s − 1.52e5·20-s + 1.55e6·21-s − 2.34e5·22-s + 7.79e5·23-s + 5.61e5·24-s − 1.59e6·25-s + 8.89e5·26-s − 2.82e6·27-s + 2.90e6·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.976·3-s + 1/2·4-s − 0.425·5-s + 0.690·6-s + 1.78·7-s + 0.353·8-s − 0.0464·9-s − 0.301·10-s − 0.301·11-s + 0.488·12-s + 0.540·13-s + 1.26·14-s − 0.415·15-s + 1/4·16-s + 1.22·17-s − 0.0328·18-s − 0.767·19-s − 0.212·20-s + 1.74·21-s − 0.213·22-s + 0.580·23-s + 0.345·24-s − 0.818·25-s + 0.381·26-s − 1.02·27-s + 0.893·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(3.777404390\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.777404390\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - p^{4} T \) |
| 11 | \( 1 + p^{4} T \) |
good | 3 | \( 1 - 137 T + p^{9} T^{2} \) |
| 5 | \( 1 + 119 p T + p^{9} T^{2} \) |
| 7 | \( 1 - 1622 p T + p^{9} T^{2} \) |
| 13 | \( 1 - 55620 T + p^{9} T^{2} \) |
| 17 | \( 1 - 421550 T + p^{9} T^{2} \) |
| 19 | \( 1 + 435872 T + p^{9} T^{2} \) |
| 23 | \( 1 - 779077 T + p^{9} T^{2} \) |
| 29 | \( 1 - 1206768 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7626195 T + p^{9} T^{2} \) |
| 37 | \( 1 + 19473681 T + p^{9} T^{2} \) |
| 41 | \( 1 + 9906168 T + p^{9} T^{2} \) |
| 43 | \( 1 + 20662730 T + p^{9} T^{2} \) |
| 47 | \( 1 - 2751600 T + p^{9} T^{2} \) |
| 53 | \( 1 - 78527114 T + p^{9} T^{2} \) |
| 59 | \( 1 + 105727893 T + p^{9} T^{2} \) |
| 61 | \( 1 + 24639860 T + p^{9} T^{2} \) |
| 67 | \( 1 + 94817369 T + p^{9} T^{2} \) |
| 71 | \( 1 - 338924343 T + p^{9} T^{2} \) |
| 73 | \( 1 - 10287972 T + p^{9} T^{2} \) |
| 79 | \( 1 - 556386626 T + p^{9} T^{2} \) |
| 83 | \( 1 - 22479190 T + p^{9} T^{2} \) |
| 89 | \( 1 + 213504377 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1512644953 T + p^{9} T^{2} \) |
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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
−15.30048706081349747751014004302, −14.56286022240412711029392365441, −13.66224107692048883566376965352, −11.98491091928900548819855946054, −10.81426654216504142344227045614, −8.562579199117801393742967285805, −7.64685484719148717564965556224, −5.25961820130201723537348793567, −3.63057057222498432959683611351, −1.84086124668774398099564633120,
1.84086124668774398099564633120, 3.63057057222498432959683611351, 5.25961820130201723537348793567, 7.64685484719148717564965556224, 8.562579199117801393742967285805, 10.81426654216504142344227045614, 11.98491091928900548819855946054, 13.66224107692048883566376965352, 14.56286022240412711029392365441, 15.30048706081349747751014004302