| L(s) = 1 | − 2-s − 1.79·3-s + 4-s − 1.46·5-s + 1.79·6-s + 2.25·7-s − 8-s + 0.228·9-s + 1.46·10-s + 0.117·11-s − 1.79·12-s − 6.28·13-s − 2.25·14-s + 2.63·15-s + 16-s − 1.43·17-s − 0.228·18-s − 1.93·19-s − 1.46·20-s − 4.05·21-s − 0.117·22-s − 6.13·23-s + 1.79·24-s − 2.85·25-s + 6.28·26-s + 4.97·27-s + 2.25·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.03·3-s + 0.5·4-s − 0.654·5-s + 0.733·6-s + 0.852·7-s − 0.353·8-s + 0.0761·9-s + 0.463·10-s + 0.0353·11-s − 0.518·12-s − 1.74·13-s − 0.602·14-s + 0.679·15-s + 0.250·16-s − 0.347·17-s − 0.0538·18-s − 0.444·19-s − 0.327·20-s − 0.884·21-s − 0.0250·22-s − 1.27·23-s + 0.366·24-s − 0.571·25-s + 1.23·26-s + 0.958·27-s + 0.426·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3026451107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3026451107\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 - 2.25T + 7T^{2} \) |
| 11 | \( 1 - 0.117T + 11T^{2} \) |
| 13 | \( 1 + 6.28T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 + 2.84T + 29T^{2} \) |
| 31 | \( 1 - 1.89T + 31T^{2} \) |
| 37 | \( 1 + 3.86T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 - 6.00T + 43T^{2} \) |
| 47 | \( 1 - 5.56T + 47T^{2} \) |
| 53 | \( 1 - 4.53T + 53T^{2} \) |
| 59 | \( 1 - 6.29T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 + 2.67T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 5.23T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.368781252909198617959612175586, −7.62297386415575724351917751690, −7.23719805397743101615736365714, −6.23081807648960026281726839326, −5.56112669084596754559209468038, −4.73569591920189558331817305394, −4.08767141663435029733818289086, −2.68394858178331084623714880490, −1.79568263929196899052515333349, −0.35612860631223642229336100529,
0.35612860631223642229336100529, 1.79568263929196899052515333349, 2.68394858178331084623714880490, 4.08767141663435029733818289086, 4.73569591920189558331817305394, 5.56112669084596754559209468038, 6.23081807648960026281726839326, 7.23719805397743101615736365714, 7.62297386415575724351917751690, 8.368781252909198617959612175586
