| L(s) = 1 | − 1.57·2-s − 1.61·3-s + 0.472·4-s + 5-s + 2.53·6-s + 4.39·7-s + 2.40·8-s − 0.404·9-s − 1.57·10-s + 4.22·11-s − 0.760·12-s − 3.23·13-s − 6.91·14-s − 1.61·15-s − 4.72·16-s + 3.49·17-s + 0.635·18-s + 2.15·19-s + 0.472·20-s − 7.08·21-s − 6.64·22-s − 3.26·23-s − 3.87·24-s + 25-s + 5.08·26-s + 5.48·27-s + 2.07·28-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.930·3-s + 0.236·4-s + 0.447·5-s + 1.03·6-s + 1.66·7-s + 0.849·8-s − 0.134·9-s − 0.497·10-s + 1.27·11-s − 0.219·12-s − 0.896·13-s − 1.84·14-s − 0.415·15-s − 1.18·16-s + 0.847·17-s + 0.149·18-s + 0.493·19-s + 0.105·20-s − 1.54·21-s − 1.41·22-s − 0.681·23-s − 0.790·24-s + 0.200·25-s + 0.997·26-s + 1.05·27-s + 0.392·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7662953899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7662953899\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 3 | \( 1 + 1.61T + 3T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 3.49T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 + 3.26T + 23T^{2} \) |
| 29 | \( 1 + 6.69T + 29T^{2} \) |
| 31 | \( 1 + 3.31T + 31T^{2} \) |
| 37 | \( 1 - 3.75T + 37T^{2} \) |
| 41 | \( 1 + 8.05T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 9.72T + 47T^{2} \) |
| 53 | \( 1 - 6.01T + 53T^{2} \) |
| 59 | \( 1 + 6.30T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 8.24T + 79T^{2} \) |
| 83 | \( 1 + 0.530T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−10.34359547185969119429827380867, −9.446501353160316340510569389132, −8.775343477371265055635819392124, −7.79379151495426390641615044165, −7.16601093383977630776881782638, −5.81662524636022121732487620395, −5.12604615847745949190466933691, −4.14534515487003161198848490809, −1.99861088658288166244862729885, −0.965158308209562193581431622674,
0.965158308209562193581431622674, 1.99861088658288166244862729885, 4.14534515487003161198848490809, 5.12604615847745949190466933691, 5.81662524636022121732487620395, 7.16601093383977630776881782638, 7.79379151495426390641615044165, 8.775343477371265055635819392124, 9.446501353160316340510569389132, 10.34359547185969119429827380867
