L(s) = 1 | + 2-s + 0.626·3-s + 4-s + 2.82·5-s + 0.626·6-s − 7-s + 8-s − 2.60·9-s + 2.82·10-s − 3.09·11-s + 0.626·12-s − 0.0836·13-s − 14-s + 1.77·15-s + 16-s + 1.78·17-s − 2.60·18-s + 2.62·19-s + 2.82·20-s − 0.626·21-s − 3.09·22-s + 0.626·24-s + 2.98·25-s − 0.0836·26-s − 3.51·27-s − 28-s + 0.800·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.361·3-s + 0.5·4-s + 1.26·5-s + 0.255·6-s − 0.377·7-s + 0.353·8-s − 0.869·9-s + 0.893·10-s − 0.933·11-s + 0.180·12-s − 0.0231·13-s − 0.267·14-s + 0.457·15-s + 0.250·16-s + 0.432·17-s − 0.614·18-s + 0.601·19-s + 0.631·20-s − 0.136·21-s − 0.659·22-s + 0.127·24-s + 0.597·25-s − 0.0163·26-s − 0.676·27-s − 0.188·28-s + 0.148·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.295721259\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.295721259\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 0.626T + 3T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 3.09T + 11T^{2} \) |
| 13 | \( 1 + 0.0836T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 29 | \( 1 - 0.800T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 + 0.184T + 37T^{2} \) |
| 41 | \( 1 + 0.0855T + 41T^{2} \) |
| 43 | \( 1 - 12.7T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 8.62T + 53T^{2} \) |
| 59 | \( 1 + 9.16T + 59T^{2} \) |
| 61 | \( 1 - 2.35T + 61T^{2} \) |
| 67 | \( 1 - 4.31T + 67T^{2} \) |
| 71 | \( 1 + 5.34T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 - 2.88T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 + 7.98T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−7.80395494069394480970955546299, −7.16447746381900355735663180537, −6.18416576840639893691887727639, −5.76716644096529885289727473720, −5.30055848792354573535982657337, −4.38682252043858008242040184603, −3.36074734503106447974877441249, −2.63286782521681986406788706662, −2.26149005150193701962963716935, −0.914456414063823283551203600431,
0.914456414063823283551203600431, 2.26149005150193701962963716935, 2.63286782521681986406788706662, 3.36074734503106447974877441249, 4.38682252043858008242040184603, 5.30055848792354573535982657337, 5.76716644096529885289727473720, 6.18416576840639893691887727639, 7.16447746381900355735663180537, 7.80395494069394480970955546299