| L(s) = 1 | + 2-s + 3-s + 4-s − 3.24·5-s + 6-s − 0.307·7-s + 8-s + 9-s − 3.24·10-s + 11-s + 12-s − 6.04·13-s − 0.307·14-s − 3.24·15-s + 16-s + 0.109·17-s + 18-s + 5.76·19-s − 3.24·20-s − 0.307·21-s + 22-s + 4.65·23-s + 24-s + 5.54·25-s − 6.04·26-s + 27-s − 0.307·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.45·5-s + 0.408·6-s − 0.116·7-s + 0.353·8-s + 0.333·9-s − 1.02·10-s + 0.301·11-s + 0.288·12-s − 1.67·13-s − 0.0823·14-s − 0.838·15-s + 0.250·16-s + 0.0266·17-s + 0.235·18-s + 1.32·19-s − 0.726·20-s − 0.0672·21-s + 0.213·22-s + 0.970·23-s + 0.204·24-s + 1.10·25-s − 1.18·26-s + 0.192·27-s − 0.0582·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 + 3.24T + 5T^{2} \) |
| 7 | \( 1 + 0.307T + 7T^{2} \) |
| 13 | \( 1 + 6.04T + 13T^{2} \) |
| 17 | \( 1 - 0.109T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 + 2.02T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 3.27T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 + 8.85T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 + 5.30T + 59T^{2} \) |
| 67 | \( 1 + 7.89T + 67T^{2} \) |
| 71 | \( 1 + 6.02T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 5.18T + 97T^{2} \) |
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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
−7.83305860422100404632770230747, −7.19757600715071660397858745182, −7.07315891504790421048003271421, −5.63399875599646044849193681573, −4.87973141913271240397290949372, −4.22969452454331069748312492958, −3.34641588043353482161704764630, −2.91977078001081960549397529989, −1.61191960021691067799328172856, 0,
1.61191960021691067799328172856, 2.91977078001081960549397529989, 3.34641588043353482161704764630, 4.22969452454331069748312492958, 4.87973141913271240397290949372, 5.63399875599646044849193681573, 7.07315891504790421048003271421, 7.19757600715071660397858745182, 7.83305860422100404632770230747
