| L(s) = 1 | − 0.181·2-s + 3.35·3-s − 1.96·4-s − 0.608·6-s − 3.58·7-s + 0.718·8-s + 8.27·9-s − 2.93·11-s − 6.60·12-s + 1.87·13-s + 0.649·14-s + 3.80·16-s − 4.52·17-s − 1.50·18-s − 4.50·19-s − 12.0·21-s + 0.531·22-s + 3.94·23-s + 2.41·24-s − 0.339·26-s + 17.7·27-s + 7.04·28-s + 1.37·29-s − 5.03·31-s − 2.12·32-s − 9.85·33-s + 0.819·34-s + ⋯ |
| L(s) = 1 | − 0.128·2-s + 1.93·3-s − 0.983·4-s − 0.248·6-s − 1.35·7-s + 0.254·8-s + 2.75·9-s − 0.884·11-s − 1.90·12-s + 0.520·13-s + 0.173·14-s + 0.951·16-s − 1.09·17-s − 0.353·18-s − 1.03·19-s − 2.62·21-s + 0.113·22-s + 0.821·23-s + 0.492·24-s − 0.0666·26-s + 3.41·27-s + 1.33·28-s + 0.254·29-s − 0.903·31-s − 0.376·32-s − 1.71·33-s + 0.140·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 0.181T + 2T^{2} \) |
| 3 | \( 1 - 3.35T + 3T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + 2.93T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 4.52T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 - 0.0281T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 - 5.99T + 43T^{2} \) |
| 47 | \( 1 + 4.69T + 47T^{2} \) |
| 53 | \( 1 + 5.15T + 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 0.184T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 8.43T + 97T^{2} \) |
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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
−8.118074923335297667382932291120, −7.39873314193606196467507705644, −6.70862935073504657488274726359, −5.74885390981082548686658966197, −4.42571064863944287493441372519, −4.15642550366991716945603223030, −3.09688353700890027695804346103, −2.76388470779156435070380465602, −1.55429673717939368883326489232, 0,
1.55429673717939368883326489232, 2.76388470779156435070380465602, 3.09688353700890027695804346103, 4.15642550366991716945603223030, 4.42571064863944287493441372519, 5.74885390981082548686658966197, 6.70862935073504657488274726359, 7.39873314193606196467507705644, 8.118074923335297667382932291120
