| L(s) = 1 | + 1.41·2-s + 2.28·5-s + 7-s − 2.82·8-s + 3.23·10-s + 4.57·11-s + 13-s + 1.41·14-s − 4.00·16-s − 5.99·17-s − 6.70·19-s + 6.47·22-s − 8.27·23-s + 0.236·25-s + 1.41·26-s − 6.86·29-s − 7.23·31-s − 8.47·34-s + 2.28·35-s − 9·37-s − 9.48·38-s − 6.47·40-s + 1.95·41-s + 8.47·43-s + ⋯ |
| L(s) = 1 | + 1.00·2-s + 1.02·5-s + 0.377·7-s − 0.999·8-s + 1.02·10-s + 1.37·11-s + 0.277·13-s + 0.377·14-s − 1.00·16-s − 1.45·17-s − 1.53·19-s + 1.37·22-s − 1.72·23-s + 0.0472·25-s + 0.277·26-s − 1.27·29-s − 1.29·31-s − 1.45·34-s + 0.386·35-s − 1.47·37-s − 1.53·38-s − 1.02·40-s + 0.305·41-s + 1.29·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + 5.99T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 8.27T + 23T^{2} \) |
| 29 | \( 1 + 6.86T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 9T + 37T^{2} \) |
| 41 | \( 1 - 1.95T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 2.70T + 61T^{2} \) |
| 67 | \( 1 + 9T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 5.76T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 4.57T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−7.60391130002162163100977181456, −6.61112458458015978884526835354, −6.08370165750614793949441245761, −5.74902668634923664404564675240, −4.66493882370942465929523624825, −4.12790372617975540363236255456, −3.56902657046808546932310118007, −2.13186683609923161175537177628, −1.84778391424332204552058528485, 0,
1.84778391424332204552058528485, 2.13186683609923161175537177628, 3.56902657046808546932310118007, 4.12790372617975540363236255456, 4.66493882370942465929523624825, 5.74902668634923664404564675240, 6.08370165750614793949441245761, 6.61112458458015978884526835354, 7.60391130002162163100977181456
