| L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 2·5-s + 2·6-s − 4·7-s + 9-s + 4·10-s − 6·11-s − 2·12-s − 4·13-s + 8·14-s + 2·15-s − 4·16-s − 6·17-s − 2·18-s − 5·19-s − 4·20-s + 4·21-s + 12·22-s − 7·23-s − 25-s + 8·26-s − 27-s − 8·28-s − 3·29-s − 4·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 1.26·10-s − 1.80·11-s − 0.577·12-s − 1.10·13-s + 2.13·14-s + 0.516·15-s − 16-s − 1.45·17-s − 0.471·18-s − 1.14·19-s − 0.894·20-s + 0.872·21-s + 2.55·22-s − 1.45·23-s − 1/5·25-s + 1.56·26-s − 0.192·27-s − 1.51·28-s − 0.557·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 137 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.18501446085806, −15.75406351389765, −15.45641615454712, −15.07884537596419, −13.71549846974440, −13.46906970243140, −12.78425933460704, −12.31022506523484, −11.80276650617475, −11.05870739838374, −10.45453445615764, −10.25923580265600, −9.751529165039881, −9.037672206771741, −8.426465404939810, −7.974125768445757, −7.267022050480415, −6.947274451576671, −6.314615871584723, −5.524618447814622, −4.723130401893151, −4.159961221209172, −3.285541220496252, −2.389420157701567, −1.901003939307167, 0, 0, 0,
1.901003939307167, 2.389420157701567, 3.285541220496252, 4.159961221209172, 4.723130401893151, 5.524618447814622, 6.314615871584723, 6.947274451576671, 7.267022050480415, 7.974125768445757, 8.426465404939810, 9.037672206771741, 9.751529165039881, 10.25923580265600, 10.45453445615764, 11.05870739838374, 11.80276650617475, 12.31022506523484, 12.78425933460704, 13.46906970243140, 13.71549846974440, 15.07884537596419, 15.45641615454712, 15.75406351389765, 16.18501446085806
