| L(s) = 1 | − 1.14·2-s − 0.245·3-s − 0.690·4-s − 0.515·5-s + 0.280·6-s + 3.83·7-s + 3.07·8-s − 2.93·9-s + 0.590·10-s + 3.31·11-s + 0.169·12-s − 5.60·13-s − 4.38·14-s + 0.126·15-s − 2.14·16-s + 2.67·17-s + 3.36·18-s − 7.74·19-s + 0.356·20-s − 0.939·21-s − 3.79·22-s − 5.95·23-s − 0.754·24-s − 4.73·25-s + 6.41·26-s + 1.45·27-s − 2.64·28-s + ⋯ |
| L(s) = 1 | − 0.809·2-s − 0.141·3-s − 0.345·4-s − 0.230·5-s + 0.114·6-s + 1.44·7-s + 1.08·8-s − 0.979·9-s + 0.186·10-s + 1.00·11-s + 0.0488·12-s − 1.55·13-s − 1.17·14-s + 0.0326·15-s − 0.535·16-s + 0.648·17-s + 0.792·18-s − 1.77·19-s + 0.0796·20-s − 0.205·21-s − 0.809·22-s − 1.24·23-s − 0.154·24-s − 0.946·25-s + 1.25·26-s + 0.280·27-s − 0.500·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5281400212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5281400212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 1.14T + 2T^{2} \) |
| 3 | \( 1 + 0.245T + 3T^{2} \) |
| 5 | \( 1 + 0.515T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 13 | \( 1 + 5.60T + 13T^{2} \) |
| 17 | \( 1 - 2.67T + 17T^{2} \) |
| 19 | \( 1 + 7.74T + 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 - 5.15T + 31T^{2} \) |
| 37 | \( 1 + 4.84T + 37T^{2} \) |
| 41 | \( 1 + 8.21T + 41T^{2} \) |
| 43 | \( 1 + 3.85T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 + 3.76T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 - 1.67T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 7.49T + 71T^{2} \) |
| 73 | \( 1 - 4.95T + 73T^{2} \) |
| 79 | \( 1 - 3.32T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 2.77T + 89T^{2} \) |
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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
−8.020532217502640020682949786596, −7.29137078271987051033317000218, −6.44270214974451519638582521121, −5.52537177993204186241736344304, −4.93613308523144803115670986065, −4.28267673358841918478267567508, −3.61560862072804042422572084866, −2.10591834241743220704882542015, −1.78827339847636058825482386198, −0.39159691840061129860123635345,
0.39159691840061129860123635345, 1.78827339847636058825482386198, 2.10591834241743220704882542015, 3.61560862072804042422572084866, 4.28267673358841918478267567508, 4.93613308523144803115670986065, 5.52537177993204186241736344304, 6.44270214974451519638582521121, 7.29137078271987051033317000218, 8.020532217502640020682949786596
