| L(s) = 1 | − 2-s − 1.82·3-s + 4-s − 3.80·5-s + 1.82·6-s − 3.54·7-s − 8-s + 0.315·9-s + 3.80·10-s − 4.23·11-s − 1.82·12-s + 1.93·13-s + 3.54·14-s + 6.92·15-s + 16-s − 7.25·17-s − 0.315·18-s + 1.65·19-s − 3.80·20-s + 6.45·21-s + 4.23·22-s + 2.77·23-s + 1.82·24-s + 9.44·25-s − 1.93·26-s + 4.88·27-s − 3.54·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.05·3-s + 0.5·4-s − 1.69·5-s + 0.743·6-s − 1.33·7-s − 0.353·8-s + 0.105·9-s + 1.20·10-s − 1.27·11-s − 0.525·12-s + 0.535·13-s + 0.947·14-s + 1.78·15-s + 0.250·16-s − 1.75·17-s − 0.0744·18-s + 0.379·19-s − 0.849·20-s + 1.40·21-s + 0.903·22-s + 0.579·23-s + 0.371·24-s + 1.88·25-s − 0.378·26-s + 0.940·27-s − 0.669·28-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 | \( 1 + 1.82T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 1.93T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 - 2.77T + 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 + 9.37T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 - 7.72T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 - 2.77T + 47T^{2} \) |
| 53 | \( 1 + 8.53T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 9.05T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 + 3.33T + 97T^{2} \) |
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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
−7.947218529264030819534182516361, −7.41849190131275264824858604583, −6.63777364396543384269564880493, −6.12919825383342650828542549011, −5.11623307771006073486135695495, −4.29093681380465449379583510807, −3.31977023282282205460793435115, −2.60791660910353681225428941729, −0.66056229024249714485810315341, 0,
0.66056229024249714485810315341, 2.60791660910353681225428941729, 3.31977023282282205460793435115, 4.29093681380465449379583510807, 5.11623307771006073486135695495, 6.12919825383342650828542549011, 6.63777364396543384269564880493, 7.41849190131275264824858604583, 7.947218529264030819534182516361
