| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s − 2·9-s − 10-s + 11-s − 12-s − 6·13-s − 14-s − 15-s + 16-s − 3·17-s + 2·18-s − 6·19-s + 20-s − 21-s − 22-s + 24-s + 25-s + 6·26-s + 5·27-s + 28-s − 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.962·27-s + 0.188·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.38543391955788, −14.73928940972760, −14.58571511037195, −14.00022491845655, −13.15703887215498, −12.68497166197313, −12.10215164505543, −11.73117184084003, −10.95153047155392, −10.80005790654181, −10.14496376157643, −9.527983366360706, −9.009781343030200, −8.557235886863663, −7.946668526358157, −7.189224537324276, −6.788196125197934, −6.220740178294629, −5.502681415961507, −5.062471782091386, −4.423034986928794, −3.537471831642021, −2.602021195568924, −2.126574748488043, −1.453794826294557, 0, 0,
1.453794826294557, 2.126574748488043, 2.602021195568924, 3.537471831642021, 4.423034986928794, 5.062471782091386, 5.502681415961507, 6.220740178294629, 6.788196125197934, 7.189224537324276, 7.946668526358157, 8.557235886863663, 9.009781343030200, 9.527983366360706, 10.14496376157643, 10.80005790654181, 10.95153047155392, 11.73117184084003, 12.10215164505543, 12.68497166197313, 13.15703887215498, 14.00022491845655, 14.58571511037195, 14.73928940972760, 15.38543391955788
