| L(s) = 1 | − 2.69·2-s + 3-s + 5.25·4-s − 3.14·5-s − 2.69·6-s + 0.0430·7-s − 8.76·8-s + 9-s + 8.47·10-s − 3.97·11-s + 5.25·12-s + 0.212·13-s − 0.115·14-s − 3.14·15-s + 13.0·16-s − 6.24·17-s − 2.69·18-s − 1.70·19-s − 16.5·20-s + 0.0430·21-s + 10.6·22-s − 1.61·23-s − 8.76·24-s + 4.90·25-s − 0.573·26-s + 27-s + 0.226·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 0.577·3-s + 2.62·4-s − 1.40·5-s − 1.09·6-s + 0.0162·7-s − 3.09·8-s + 0.333·9-s + 2.68·10-s − 1.19·11-s + 1.51·12-s + 0.0590·13-s − 0.0309·14-s − 0.812·15-s + 3.27·16-s − 1.51·17-s − 0.634·18-s − 0.390·19-s − 3.69·20-s + 0.00939·21-s + 2.27·22-s − 0.337·23-s − 1.78·24-s + 0.981·25-s − 0.112·26-s + 0.192·27-s + 0.0427·28-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
| \(L(1)\) |
\(\approx\) |
\(0.03351023141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03351023141\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.69T + 2T^{2} \) |
| 5 | \( 1 + 3.14T + 5T^{2} \) |
| 7 | \( 1 - 0.0430T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 13 | \( 1 - 0.212T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 + 8.54T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 7.20T + 37T^{2} \) |
| 41 | \( 1 - 6.68T + 41T^{2} \) |
| 43 | \( 1 + 8.46T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + 8.24T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 + 0.662T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 1.48T + 89T^{2} \) |
| 97 | \( 1 - 0.764T + 97T^{2} \) |
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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
−8.111361792126498363829744481905, −7.34540703957547095101188657165, −7.06568642160034785399639665099, −6.15324989286596748405191976861, −5.06732820835878483135092021708, −3.98277482777067172583316957676, −3.25599391194056709826985495005, −2.35740481698565191103235660190, −1.69204213195853549371513167135, −0.11108960758647804363133230604,
0.11108960758647804363133230604, 1.69204213195853549371513167135, 2.35740481698565191103235660190, 3.25599391194056709826985495005, 3.98277482777067172583316957676, 5.06732820835878483135092021708, 6.15324989286596748405191976861, 7.06568642160034785399639665099, 7.34540703957547095101188657165, 8.111361792126498363829744481905
