| L(s) = 1 | + 0.268·2-s − 2.54·3-s − 1.92·4-s − 0.0127·5-s − 0.683·6-s + 1.14·7-s − 1.05·8-s + 3.49·9-s − 0.00340·10-s − 3.53·11-s + 4.91·12-s + 5.69·13-s + 0.305·14-s + 0.0323·15-s + 3.57·16-s − 2.24·17-s + 0.937·18-s + 2.38·19-s + 0.0244·20-s − 2.90·21-s − 0.948·22-s + 8.30·23-s + 2.68·24-s − 4.99·25-s + 1.52·26-s − 1.26·27-s − 2.19·28-s + ⋯ |
| L(s) = 1 | + 0.189·2-s − 1.47·3-s − 0.964·4-s − 0.00568·5-s − 0.279·6-s + 0.431·7-s − 0.372·8-s + 1.16·9-s − 0.00107·10-s − 1.06·11-s + 1.41·12-s + 1.57·13-s + 0.0817·14-s + 0.00836·15-s + 0.893·16-s − 0.545·17-s + 0.221·18-s + 0.548·19-s + 0.00547·20-s − 0.634·21-s − 0.202·22-s + 1.73·23-s + 0.548·24-s − 0.999·25-s + 0.299·26-s − 0.244·27-s − 0.415·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1247 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1247 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 0.268T + 2T^{2} \) |
| 3 | \( 1 + 2.54T + 3T^{2} \) |
| 5 | \( 1 + 0.0127T + 5T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 3.53T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 8.30T + 23T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 + 1.74T + 41T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 - 3.30T + 59T^{2} \) |
| 61 | \( 1 + 9.97T + 61T^{2} \) |
| 67 | \( 1 - 9.68T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 0.605T + 83T^{2} \) |
| 89 | \( 1 + 6.11T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−9.295287658673927464371234358463, −8.549175956328026464106440660483, −7.62870366851729367846089927412, −6.56940255821513340642042705081, −5.58535740538611173399211579390, −5.26148387079755589419579374429, −4.36295021922158551475390056373, −3.29683937108007530910112736928, −1.33468956894289870737997698775, 0,
1.33468956894289870737997698775, 3.29683937108007530910112736928, 4.36295021922158551475390056373, 5.26148387079755589419579374429, 5.58535740538611173399211579390, 6.56940255821513340642042705081, 7.62870366851729367846089927412, 8.549175956328026464106440660483, 9.295287658673927464371234358463
