L(s) = 1 | + 2-s + 1.43·3-s + 4-s − 5-s + 1.43·6-s + 3.08·7-s + 8-s − 0.950·9-s − 10-s − 6.46·11-s + 1.43·12-s + 3.95·13-s + 3.08·14-s − 1.43·15-s + 16-s − 3.43·17-s − 0.950·18-s + 3.08·19-s − 20-s + 4.41·21-s − 6.46·22-s − 23-s + 1.43·24-s + 25-s + 3.95·26-s − 5.65·27-s + 3.08·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.826·3-s + 0.5·4-s − 0.447·5-s + 0.584·6-s + 1.16·7-s + 0.353·8-s − 0.316·9-s − 0.316·10-s − 1.95·11-s + 0.413·12-s + 1.09·13-s + 0.825·14-s − 0.369·15-s + 0.250·16-s − 0.832·17-s − 0.224·18-s + 0.708·19-s − 0.223·20-s + 0.964·21-s − 1.37·22-s − 0.208·23-s + 0.292·24-s + 0.200·25-s + 0.774·26-s − 1.08·27-s + 0.583·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.160604022\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160604022\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 1.43T + 3T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 + 6.46T + 11T^{2} \) |
| 13 | \( 1 - 3.95T + 13T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 - 3.08T + 19T^{2} \) |
| 29 | \( 1 - 0.863T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 + 7.03T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 3.90T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.86T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.56T + 71T^{2} \) |
| 73 | \( 1 - 5.90T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 9.03T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−12.33475975770652326267792039030, −11.09848412659422462184542358089, −10.76551686819435132078406800875, −9.000035973415752322031265082013, −8.063154177754476700386349911353, −7.52358087496945188915114426824, −5.76389467487929549441661298382, −4.78597473354651412117779600274, −3.44779765652501183131218920925, −2.22861792377438448870645236794,
2.22861792377438448870645236794, 3.44779765652501183131218920925, 4.78597473354651412117779600274, 5.76389467487929549441661298382, 7.52358087496945188915114426824, 8.063154177754476700386349911353, 9.000035973415752322031265082013, 10.76551686819435132078406800875, 11.09848412659422462184542358089, 12.33475975770652326267792039030