| L(s) = 1 | + 0.246·2-s + 3-s − 1.93·4-s − 0.318·5-s + 0.246·6-s + 7-s − 0.969·8-s + 9-s − 0.0783·10-s + 2.79·11-s − 1.93·12-s − 6.49·13-s + 0.246·14-s − 0.318·15-s + 3.63·16-s − 1.33·17-s + 0.246·18-s + 4.36·19-s + 0.617·20-s + 21-s + 0.689·22-s − 5.29·23-s − 0.969·24-s − 4.89·25-s − 1.60·26-s + 27-s − 1.93·28-s + ⋯ |
| L(s) = 1 | + 0.174·2-s + 0.577·3-s − 0.969·4-s − 0.142·5-s + 0.100·6-s + 0.377·7-s − 0.342·8-s + 0.333·9-s − 0.0247·10-s + 0.844·11-s − 0.559·12-s − 1.80·13-s + 0.0658·14-s − 0.0821·15-s + 0.909·16-s − 0.322·17-s + 0.0580·18-s + 1.00·19-s + 0.137·20-s + 0.218·21-s + 0.146·22-s − 1.10·23-s − 0.197·24-s − 0.979·25-s − 0.313·26-s + 0.192·27-s − 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.246T + 2T^{2} \) |
| 5 | \( 1 + 0.318T + 5T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 + 1.33T + 17T^{2} \) |
| 19 | \( 1 - 4.36T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 - 8.04T + 43T^{2} \) |
| 47 | \( 1 + 7.16T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 + 5.02T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 - 0.549T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 1.49T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7.20T + 97T^{2} \) |
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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
−8.546921341492330181885799681287, −7.63934513967805420733838942281, −7.30949400107722559280277794780, −5.99768189424270587415592405735, −5.20976027816489578293496842880, −4.36664316523829153446406684357, −3.82083415866875313715198795380, −2.73082487966252966850486220613, −1.59943812692071389512082881175, 0,
1.59943812692071389512082881175, 2.73082487966252966850486220613, 3.82083415866875313715198795380, 4.36664316523829153446406684357, 5.20976027816489578293496842880, 5.99768189424270587415592405735, 7.30949400107722559280277794780, 7.63934513967805420733838942281, 8.546921341492330181885799681287
