| L(s) = 1 | + 2-s + 2.19·3-s + 4-s − 2.55·5-s + 2.19·6-s + 1.36·7-s + 8-s + 1.81·9-s − 2.55·10-s − 1.75·11-s + 2.19·12-s + 5.19·13-s + 1.36·14-s − 5.61·15-s + 16-s + 4.61·17-s + 1.81·18-s − 2.86·19-s − 2.55·20-s + 3.00·21-s − 1.75·22-s + 0.549·23-s + 2.19·24-s + 1.54·25-s + 5.19·26-s − 2.59·27-s + 1.36·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.26·3-s + 0.5·4-s − 1.14·5-s + 0.896·6-s + 0.517·7-s + 0.353·8-s + 0.605·9-s − 0.808·10-s − 0.529·11-s + 0.633·12-s + 1.43·13-s + 0.365·14-s − 1.44·15-s + 0.250·16-s + 1.11·17-s + 0.428·18-s − 0.656·19-s − 0.572·20-s + 0.655·21-s − 0.374·22-s + 0.114·23-s + 0.448·24-s + 0.308·25-s + 1.01·26-s − 0.499·27-s + 0.258·28-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
| \(L(1)\) |
\(\approx\) |
\(4.403791261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.403791261\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 13 | \( 1 - 5.19T + 13T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 0.549T + 23T^{2} \) |
| 29 | \( 1 - 8.02T + 29T^{2} \) |
| 31 | \( 1 - 0.211T + 31T^{2} \) |
| 37 | \( 1 - 0.867T + 37T^{2} \) |
| 41 | \( 1 - 7.57T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 - 5.10T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 4.88T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 - 5.86T + 79T^{2} \) |
| 83 | \( 1 - 2.26T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 3.00T + 97T^{2} \) |
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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
−8.291456443861399352927764008959, −7.902327520798629501511327869937, −7.19798850657175928972987749730, −6.21081476803046245073781064411, −5.35444688449814813554943978596, −4.33879876346975742537601545169, −3.79596093715243154903918018513, −3.14728993445671536965703543762, −2.32875110196867380058906269891, −1.07969243691774713822153316222,
1.07969243691774713822153316222, 2.32875110196867380058906269891, 3.14728993445671536965703543762, 3.79596093715243154903918018513, 4.33879876346975742537601545169, 5.35444688449814813554943978596, 6.21081476803046245073781064411, 7.19798850657175928972987749730, 7.902327520798629501511327869937, 8.291456443861399352927764008959
