| L(s) = 1 | + 2.23·2-s − 3.23·3-s + 3.00·4-s − 7.23·6-s − 7-s + 2.23·8-s + 7.47·9-s − 11-s − 9.70·12-s + 1.23·13-s − 2.23·14-s − 0.999·16-s − 1.23·17-s + 16.7·18-s − 2.47·19-s + 3.23·21-s − 2.23·22-s + 6.47·23-s − 7.23·24-s + 2.76·26-s − 14.4·27-s − 3.00·28-s − 0.472·29-s − 7.23·31-s − 6.70·32-s + 3.23·33-s − 2.76·34-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 1.86·3-s + 1.50·4-s − 2.95·6-s − 0.377·7-s + 0.790·8-s + 2.49·9-s − 0.301·11-s − 2.80·12-s + 0.342·13-s − 0.597·14-s − 0.249·16-s − 0.299·17-s + 3.93·18-s − 0.567·19-s + 0.706·21-s − 0.476·22-s + 1.34·23-s − 1.47·24-s + 0.542·26-s − 2.78·27-s − 0.566·28-s − 0.0876·29-s − 1.29·31-s − 1.18·32-s + 0.563·33-s − 0.474·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 3.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 5.52T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 - 5.23T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−8.904963126257412663652940308040, −7.45220159823844055692046140664, −6.59921279496292326707982342873, −6.35038165499467769702369914702, −5.31215301317140184313862683254, −5.03007028540882978007051644622, −4.12139268585744887520408968925, −3.21107026572275759736294232030, −1.69968158475737578446000399210, 0,
1.69968158475737578446000399210, 3.21107026572275759736294232030, 4.12139268585744887520408968925, 5.03007028540882978007051644622, 5.31215301317140184313862683254, 6.35038165499467769702369914702, 6.59921279496292326707982342873, 7.45220159823844055692046140664, 8.904963126257412663652940308040
