| L(s) = 1 | − 2.61·2-s + 0.618·3-s + 4.85·4-s + 5-s − 1.61·6-s + 7-s − 7.47·8-s − 2.61·9-s − 2.61·10-s + 3.00·12-s + 3.23·13-s − 2.61·14-s + 0.618·15-s + 9.85·16-s − 8.09·17-s + 6.85·18-s − 6.23·19-s + 4.85·20-s + 0.618·21-s − 6.09·23-s − 4.61·24-s − 4·25-s − 8.47·26-s − 3.47·27-s + 4.85·28-s − 2.38·29-s − 1.61·30-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.356·3-s + 2.42·4-s + 0.447·5-s − 0.660·6-s + 0.377·7-s − 2.64·8-s − 0.872·9-s − 0.827·10-s + 0.866·12-s + 0.897·13-s − 0.699·14-s + 0.159·15-s + 2.46·16-s − 1.96·17-s + 1.61·18-s − 1.43·19-s + 1.08·20-s + 0.134·21-s − 1.26·23-s − 0.942·24-s − 0.800·25-s − 1.66·26-s − 0.668·27-s + 0.917·28-s − 0.442·29-s − 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 0.618T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 8.09T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 + 2.38T + 29T^{2} \) |
| 31 | \( 1 - 0.236T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 + 4.38T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 + 0.0901T + 59T^{2} \) |
| 61 | \( 1 + 5.38T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 + 4.90T + 71T^{2} \) |
| 73 | \( 1 + 9.76T + 73T^{2} \) |
| 79 | \( 1 - 8.61T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 0.145T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.482838818025454908029145363388, −8.876017411837362029824939682435, −8.366164039787206267798900311225, −7.61495633123750741670707522513, −6.39173109338585073287907499930, −5.97163257254285904479895135582, −4.11489933847878249489477622553, −2.48188325004143288463047994564, −1.85083387354783508676328934534, 0,
1.85083387354783508676328934534, 2.48188325004143288463047994564, 4.11489933847878249489477622553, 5.97163257254285904479895135582, 6.39173109338585073287907499930, 7.61495633123750741670707522513, 8.366164039787206267798900311225, 8.876017411837362029824939682435, 9.482838818025454908029145363388
