L(s) = 1 | − 2-s − 0.923·3-s + 4-s − 1.00·5-s + 0.923·6-s − 1.65·7-s − 8-s − 2.14·9-s + 1.00·10-s − 3.52·11-s − 0.923·12-s − 2.87·13-s + 1.65·14-s + 0.928·15-s + 16-s − 0.517·17-s + 2.14·18-s − 2.41·19-s − 1.00·20-s + 1.52·21-s + 3.52·22-s − 7.18·23-s + 0.923·24-s − 3.98·25-s + 2.87·26-s + 4.75·27-s − 1.65·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.533·3-s + 0.5·4-s − 0.449·5-s + 0.377·6-s − 0.625·7-s − 0.353·8-s − 0.715·9-s + 0.317·10-s − 1.06·11-s − 0.266·12-s − 0.796·13-s + 0.442·14-s + 0.239·15-s + 0.250·16-s − 0.125·17-s + 0.505·18-s − 0.554·19-s − 0.224·20-s + 0.333·21-s + 0.752·22-s − 1.49·23-s + 0.188·24-s − 0.797·25-s + 0.563·26-s + 0.915·27-s − 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02115975205\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02115975205\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 + 0.923T + 3T^{2} \) |
| 5 | \( 1 + 1.00T + 5T^{2} \) |
| 7 | \( 1 + 1.65T + 7T^{2} \) |
| 11 | \( 1 + 3.52T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 + 0.517T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 + 9.53T + 37T^{2} \) |
| 41 | \( 1 + 4.03T + 41T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 1.45T + 47T^{2} \) |
| 53 | \( 1 + 7.71T + 53T^{2} \) |
| 59 | \( 1 + 6.62T + 59T^{2} \) |
| 61 | \( 1 - 5.00T + 61T^{2} \) |
| 67 | \( 1 + 8.51T + 67T^{2} \) |
| 71 | \( 1 - 3.57T + 71T^{2} \) |
| 73 | \( 1 - 0.496T + 73T^{2} \) |
| 79 | \( 1 + 3.81T + 79T^{2} \) |
| 83 | \( 1 + 4.39T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.374434385205459911742885591905, −7.77388175757366504463031644218, −7.11976254085594998781472875868, −6.21808220236356313278040871216, −5.66451862127044204105371587590, −4.81597278308507150860642670652, −3.72472868104478188195482036938, −2.80748427126123209962720865659, −1.95396395408978061087424711414, −0.087578655656298897786562394226,
0.087578655656298897786562394226, 1.95396395408978061087424711414, 2.80748427126123209962720865659, 3.72472868104478188195482036938, 4.81597278308507150860642670652, 5.66451862127044204105371587590, 6.21808220236356313278040871216, 7.11976254085594998781472875868, 7.77388175757366504463031644218, 8.374434385205459911742885591905