| L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s + 3·11-s − 4·13-s − 2·14-s + 16-s + 5·17-s + 6·19-s − 3·22-s + 3·23-s + 4·26-s + 2·28-s − 3·29-s + 2·31-s − 32-s − 5·34-s − 2·37-s − 6·38-s − 8·41-s − 2·43-s + 3·44-s − 3·46-s − 3·49-s − 4·52-s − 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s + 0.904·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 1.21·17-s + 1.37·19-s − 0.639·22-s + 0.625·23-s + 0.784·26-s + 0.377·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.857·34-s − 0.328·37-s − 0.973·38-s − 1.24·41-s − 0.304·43-s + 0.452·44-s − 0.442·46-s − 3/7·49-s − 0.554·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.62497426005823, −14.19073451037071, −13.83188332038140, −13.03701321083363, −12.31568966005204, −11.99143406005974, −11.61149216594825, −11.07656783223957, −10.46615313676195, −9.810500484133597, −9.567834598372222, −9.042801125926551, −8.313129429725705, −7.881321537625454, −7.367255195471364, −6.946332156701687, −6.283826200201384, −5.484358204670921, −5.096907852877655, −4.513825730370054, −3.516284216973129, −3.168021806671369, −2.299485358053838, −1.463624786031013, −1.123543433492543, 0,
1.123543433492543, 1.463624786031013, 2.299485358053838, 3.168021806671369, 3.516284216973129, 4.513825730370054, 5.096907852877655, 5.484358204670921, 6.283826200201384, 6.946332156701687, 7.367255195471364, 7.881321537625454, 8.313129429725705, 9.042801125926551, 9.567834598372222, 9.810500484133597, 10.46615313676195, 11.07656783223957, 11.61149216594825, 11.99143406005974, 12.31568966005204, 13.03701321083363, 13.83188332038140, 14.19073451037071, 14.62497426005823
