| L(s) = 1 | − 2.21·2-s + 3-s + 2.92·4-s − 1.40·5-s − 2.21·6-s + 0.0138·7-s − 2.04·8-s + 9-s + 3.12·10-s − 6.42·11-s + 2.92·12-s − 5.42·13-s − 0.0306·14-s − 1.40·15-s − 1.30·16-s + 17-s − 2.21·18-s − 6.35·19-s − 4.11·20-s + 0.0138·21-s + 14.2·22-s + 2.20·23-s − 2.04·24-s − 3.01·25-s + 12.0·26-s + 27-s + 0.0403·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 0.577·3-s + 1.46·4-s − 0.629·5-s − 0.905·6-s + 0.00522·7-s − 0.722·8-s + 0.333·9-s + 0.987·10-s − 1.93·11-s + 0.843·12-s − 1.50·13-s − 0.00819·14-s − 0.363·15-s − 0.327·16-s + 0.242·17-s − 0.522·18-s − 1.45·19-s − 0.919·20-s + 0.00301·21-s + 3.03·22-s + 0.460·23-s − 0.417·24-s − 0.603·25-s + 2.35·26-s + 0.192·27-s + 0.00763·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1123485428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1123485428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 - 0.0138T + 7T^{2} \) |
| 11 | \( 1 + 6.42T + 11T^{2} \) |
| 13 | \( 1 + 5.42T + 13T^{2} \) |
| 19 | \( 1 + 6.35T + 19T^{2} \) |
| 23 | \( 1 - 2.20T + 23T^{2} \) |
| 29 | \( 1 + 4.83T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 - 8.05T + 37T^{2} \) |
| 41 | \( 1 - 8.62T + 41T^{2} \) |
| 43 | \( 1 + 7.48T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 9.77T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 7.61T + 67T^{2} \) |
| 71 | \( 1 - 4.93T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 5.96T + 89T^{2} \) |
| 97 | \( 1 + 8.70T + 97T^{2} \) |
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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
−7.88795223944439296665070354412, −7.59621338662260868567423147480, −6.95082319664989340399623909853, −5.95404848471055504057693582361, −4.91071046445150067706743949880, −4.35770427317872908943650586887, −3.08133157099266664485736828569, −2.45843165748972761866072877661, −1.76106122795402742083159210514, −0.19195386333542298382247736148,
0.19195386333542298382247736148, 1.76106122795402742083159210514, 2.45843165748972761866072877661, 3.08133157099266664485736828569, 4.35770427317872908943650586887, 4.91071046445150067706743949880, 5.95404848471055504057693582361, 6.95082319664989340399623909853, 7.59621338662260868567423147480, 7.88795223944439296665070354412
