| L(s) = 1 | − 2.62·2-s + 3-s + 4.88·4-s − 3.97·5-s − 2.62·6-s + 3.25·7-s − 7.55·8-s + 9-s + 10.4·10-s + 3.60·11-s + 4.88·12-s + 2.61·13-s − 8.55·14-s − 3.97·15-s + 10.0·16-s − 5.68·17-s − 2.62·18-s − 4.08·19-s − 19.4·20-s + 3.25·21-s − 9.44·22-s + 5.60·23-s − 7.55·24-s + 10.8·25-s − 6.84·26-s + 27-s + 15.9·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.577·3-s + 2.44·4-s − 1.77·5-s − 1.07·6-s + 1.23·7-s − 2.67·8-s + 0.333·9-s + 3.29·10-s + 1.08·11-s + 1.40·12-s + 0.723·13-s − 2.28·14-s − 1.02·15-s + 2.51·16-s − 1.37·17-s − 0.618·18-s − 0.936·19-s − 4.33·20-s + 0.711·21-s − 2.01·22-s + 1.16·23-s − 1.54·24-s + 2.16·25-s − 1.34·26-s + 0.192·27-s + 3.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9741232749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9741232749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 5 | \( 1 + 3.97T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 3.60T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 23 | \( 1 - 5.60T + 23T^{2} \) |
| 29 | \( 1 - 8.14T + 29T^{2} \) |
| 31 | \( 1 - 4.64T + 31T^{2} \) |
| 37 | \( 1 - 8.03T + 37T^{2} \) |
| 41 | \( 1 - 0.496T + 41T^{2} \) |
| 43 | \( 1 + 7.11T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 6.56T + 61T^{2} \) |
| 67 | \( 1 - 8.65T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 0.656T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 - 0.211T + 89T^{2} \) |
| 97 | \( 1 - 1.75T + 97T^{2} \) |
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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
−8.107910843786657415984475205612, −7.52714845797393155499659667629, −6.66891138260358479640391520486, −6.49526708893669084938007330599, −4.68012155594799966762895341433, −4.27827382004950280362807287862, −3.28389133534328199377782112908, −2.40347268531985459802335796071, −1.39156024588687663629527855973, −0.69990279198550144539438696533,
0.69990279198550144539438696533, 1.39156024588687663629527855973, 2.40347268531985459802335796071, 3.28389133534328199377782112908, 4.27827382004950280362807287862, 4.68012155594799966762895341433, 6.49526708893669084938007330599, 6.66891138260358479640391520486, 7.52714845797393155499659667629, 8.107910843786657415984475205612
