| L(s) = 1 | + 2-s − 1.77·3-s + 4-s − 2.30·5-s − 1.77·6-s + 0.622·7-s + 8-s + 0.138·9-s − 2.30·10-s + 3.36·11-s − 1.77·12-s + 1.06·13-s + 0.622·14-s + 4.08·15-s + 16-s − 5.10·17-s + 0.138·18-s − 4.45·19-s − 2.30·20-s − 1.10·21-s + 3.36·22-s + 4.96·23-s − 1.77·24-s + 0.313·25-s + 1.06·26-s + 5.06·27-s + 0.622·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.02·3-s + 0.5·4-s − 1.03·5-s − 0.723·6-s + 0.235·7-s + 0.353·8-s + 0.0460·9-s − 0.728·10-s + 1.01·11-s − 0.511·12-s + 0.294·13-s + 0.166·14-s + 1.05·15-s + 0.250·16-s − 1.23·17-s + 0.0325·18-s − 1.02·19-s − 0.515·20-s − 0.240·21-s + 0.716·22-s + 1.03·23-s − 0.361·24-s + 0.0626·25-s + 0.208·26-s + 0.975·27-s + 0.117·28-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 | \( 1 + 1.77T + 3T^{2} \) |
| 5 | \( 1 + 2.30T + 5T^{2} \) |
| 7 | \( 1 - 0.622T + 7T^{2} \) |
| 11 | \( 1 - 3.36T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 - 1.99T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 - 3.72T + 41T^{2} \) |
| 43 | \( 1 - 3.30T + 43T^{2} \) |
| 47 | \( 1 + 7.37T + 47T^{2} \) |
| 53 | \( 1 + 2.35T + 53T^{2} \) |
| 59 | \( 1 - 0.261T + 59T^{2} \) |
| 61 | \( 1 + 6.12T + 61T^{2} \) |
| 67 | \( 1 + 3.18T + 67T^{2} \) |
| 71 | \( 1 - 9.01T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 + 4.23T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 4.21T + 97T^{2} \) |
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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
−8.099722354899074281386236709488, −6.93161983274200221539861674198, −6.59608651240447770089432725659, −5.91669864337059700600443231871, −4.86876112135356409994937155917, −4.43384290520880824894308347874, −3.69585511988994549321304856143, −2.64602583177434262451147376770, −1.30441969764939962278388676754, 0,
1.30441969764939962278388676754, 2.64602583177434262451147376770, 3.69585511988994549321304856143, 4.43384290520880824894308347874, 4.86876112135356409994937155917, 5.91669864337059700600443231871, 6.59608651240447770089432725659, 6.93161983274200221539861674198, 8.099722354899074281386236709488
