| L(s) = 1 | − 0.217·3-s + 1.76·5-s + 1.63·7-s − 2.95·9-s − 4.73·11-s + 4.29·13-s − 0.383·15-s − 6.17·17-s + 3.23·19-s − 0.354·21-s − 5.20·23-s − 1.89·25-s + 1.29·27-s − 5.37·29-s + 3.65·31-s + 1.02·33-s + 2.87·35-s + 10.2·37-s − 0.933·39-s + 12.3·41-s − 4.50·43-s − 5.20·45-s + 11.0·47-s − 4.33·49-s + 1.34·51-s − 2.84·53-s − 8.34·55-s + ⋯ |
| L(s) = 1 | − 0.125·3-s + 0.788·5-s + 0.617·7-s − 0.984·9-s − 1.42·11-s + 1.19·13-s − 0.0989·15-s − 1.49·17-s + 0.742·19-s − 0.0774·21-s − 1.08·23-s − 0.378·25-s + 0.248·27-s − 0.997·29-s + 0.655·31-s + 0.179·33-s + 0.486·35-s + 1.68·37-s − 0.149·39-s + 1.92·41-s − 0.686·43-s − 0.776·45-s + 1.61·47-s − 0.619·49-s + 0.187·51-s − 0.390·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.855923265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855923265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
| good | 3 | \( 1 + 0.217T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 - 1.63T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 6.17T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 5.20T + 23T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 + 4.50T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 + 0.223T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 7.71T + 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 - 6.65T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 + 1.13T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.993230307066145936300773873579, −7.64005049895708087594073084276, −6.34936386848139256726197404004, −5.95084866905485137212321671341, −5.36098646488719307012978057465, −4.56666040021401909985474544102, −3.64061960636620159590588200598, −2.48659532301624752072255583420, −2.10942882749527395757841154690, −0.69853009255750439657647727298,
0.69853009255750439657647727298, 2.10942882749527395757841154690, 2.48659532301624752072255583420, 3.64061960636620159590588200598, 4.56666040021401909985474544102, 5.36098646488719307012978057465, 5.95084866905485137212321671341, 6.34936386848139256726197404004, 7.64005049895708087594073084276, 7.993230307066145936300773873579
