| L(s) = 1 | + 2-s − 2.67·3-s + 4-s + 3.78·5-s − 2.67·6-s + 0.625·7-s + 8-s + 4.17·9-s + 3.78·10-s + 0.853·11-s − 2.67·12-s + 2.70·13-s + 0.625·14-s − 10.1·15-s + 16-s − 5.14·17-s + 4.17·18-s + 4.43·19-s + 3.78·20-s − 1.67·21-s + 0.853·22-s + 3.96·23-s − 2.67·24-s + 9.31·25-s + 2.70·26-s − 3.15·27-s + 0.625·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.54·3-s + 0.5·4-s + 1.69·5-s − 1.09·6-s + 0.236·7-s + 0.353·8-s + 1.39·9-s + 1.19·10-s + 0.257·11-s − 0.773·12-s + 0.751·13-s + 0.167·14-s − 2.61·15-s + 0.250·16-s − 1.24·17-s + 0.984·18-s + 1.01·19-s + 0.845·20-s − 0.365·21-s + 0.182·22-s + 0.827·23-s − 0.546·24-s + 1.86·25-s + 0.531·26-s − 0.606·27-s + 0.118·28-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
| \(L(1)\) |
\(\approx\) |
\(2.843974788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.843974788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.67T + 3T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 7 | \( 1 - 0.625T + 7T^{2} \) |
| 11 | \( 1 - 0.853T + 11T^{2} \) |
| 13 | \( 1 - 2.70T + 13T^{2} \) |
| 17 | \( 1 + 5.14T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 - 3.96T + 23T^{2} \) |
| 29 | \( 1 + 1.84T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 - 8.82T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 - 7.78T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 + 5.74T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 7.72T + 79T^{2} \) |
| 83 | \( 1 + 0.592T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
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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
−8.598866105967090628647010562849, −7.18998986804464167407227485431, −6.59081932088015223005487844032, −6.15932256642067629514049978745, −5.38661254766708935646631321670, −5.08744440305102904967299373254, −4.17547539680822476873208499220, −2.90844237735435105689294381185, −1.82887866742850291958247874084, −1.01571812965613960919353835917,
1.01571812965613960919353835917, 1.82887866742850291958247874084, 2.90844237735435105689294381185, 4.17547539680822476873208499220, 5.08744440305102904967299373254, 5.38661254766708935646631321670, 6.15932256642067629514049978745, 6.59081932088015223005487844032, 7.18998986804464167407227485431, 8.598866105967090628647010562849
