| L(s) = 1 | + 2-s − 3.10·3-s + 4-s − 0.905·5-s − 3.10·6-s + 3.10·7-s + 8-s + 6.65·9-s − 0.905·10-s − 1.11·11-s − 3.10·12-s + 2.74·13-s + 3.10·14-s + 2.81·15-s + 16-s − 5.91·17-s + 6.65·18-s − 6.42·19-s − 0.905·20-s − 9.63·21-s − 1.11·22-s − 2.93·23-s − 3.10·24-s − 4.17·25-s + 2.74·26-s − 11.3·27-s + 3.10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.79·3-s + 0.5·4-s − 0.405·5-s − 1.26·6-s + 1.17·7-s + 0.353·8-s + 2.21·9-s − 0.286·10-s − 0.335·11-s − 0.897·12-s + 0.761·13-s + 0.828·14-s + 0.726·15-s + 0.250·16-s − 1.43·17-s + 1.56·18-s − 1.47·19-s − 0.202·20-s − 2.10·21-s − 0.237·22-s − 0.611·23-s − 0.634·24-s − 0.835·25-s + 0.538·26-s − 2.18·27-s + 0.585·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 0.905T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 13 | \( 1 - 2.74T + 13T^{2} \) |
| 17 | \( 1 + 5.91T + 17T^{2} \) |
| 19 | \( 1 + 6.42T + 19T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 - 2.20T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 3.60T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 + 4.79T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 8.00T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 + 11.4T + 83T^{2} \) |
| 89 | \( 1 - 5.76T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.062143678500706920068856313875, −7.01203513521322555484569340721, −6.27627393642363744489307068405, −6.00404687995479956409107825087, −4.81598613774741173961239147513, −4.61661001210084692978498863206, −3.93795579042862617645951450729, −2.31765854495768422446754766683, −1.35217409763407879848308209415, 0,
1.35217409763407879848308209415, 2.31765854495768422446754766683, 3.93795579042862617645951450729, 4.61661001210084692978498863206, 4.81598613774741173961239147513, 6.00404687995479956409107825087, 6.27627393642363744489307068405, 7.01203513521322555484569340721, 8.062143678500706920068856313875
