| L(s) = 1 | − 2-s − 0.0866·3-s + 4-s + 2.47·5-s + 0.0866·6-s + 0.719·7-s − 8-s − 2.99·9-s − 2.47·10-s − 4.02·11-s − 0.0866·12-s − 1.17·13-s − 0.719·14-s − 0.214·15-s + 16-s + 0.882·17-s + 2.99·18-s + 0.0145·19-s + 2.47·20-s − 0.0623·21-s + 4.02·22-s − 5.06·23-s + 0.0866·24-s + 1.12·25-s + 1.17·26-s + 0.519·27-s + 0.719·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0500·3-s + 0.5·4-s + 1.10·5-s + 0.0353·6-s + 0.272·7-s − 0.353·8-s − 0.997·9-s − 0.782·10-s − 1.21·11-s − 0.0250·12-s − 0.325·13-s − 0.192·14-s − 0.0553·15-s + 0.250·16-s + 0.213·17-s + 0.705·18-s + 0.00333·19-s + 0.553·20-s − 0.0136·21-s + 0.857·22-s − 1.05·23-s + 0.0176·24-s + 0.225·25-s + 0.230·26-s + 0.0998·27-s + 0.136·28-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 | \( 1 + 0.0866T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 - 0.719T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 + 1.17T + 13T^{2} \) |
| 17 | \( 1 - 0.882T + 17T^{2} \) |
| 19 | \( 1 - 0.0145T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 - 5.12T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 6.06T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 - 9.75T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 + 8.14T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 0.575T + 97T^{2} \) |
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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
−7.988481287131898570224324498054, −7.73752003758449273761123675513, −6.42318457579535788134622999756, −5.99250787225271532676698946759, −5.30585832008917775433091251645, −4.42968283429212237487152078616, −2.82160765253929360204566925796, −2.56307848062865978667675259680, −1.40602757822128973514647459473, 0,
1.40602757822128973514647459473, 2.56307848062865978667675259680, 2.82160765253929360204566925796, 4.42968283429212237487152078616, 5.30585832008917775433091251645, 5.99250787225271532676698946759, 6.42318457579535788134622999756, 7.73752003758449273761123675513, 7.988481287131898570224324498054
