L(s) = 1 | − 2-s + 2·3-s − 4-s − 5-s − 2·6-s + 7-s + 3·8-s + 9-s + 10-s + 2·11-s − 2·12-s + 2·13-s − 14-s − 2·15-s − 16-s − 2·17-s − 18-s + 4·19-s + 20-s + 2·21-s − 2·22-s + 6·24-s + 25-s − 2·26-s − 4·27-s − 28-s − 10·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s − 0.577·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.436·21-s − 0.426·22-s + 1.22·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s − 0.188·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33635 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.08459784126391, −14.74523383362685, −14.20763325188705, −13.76878611173295, −13.23760299352853, −12.90204161359496, −11.97930817091387, −11.42601761474547, −11.01277821271077, −10.34041118081979, −9.599264167652006, −9.220997802862314, −8.845864639293126, −8.374171937444584, −7.640797830546499, −7.565660146619165, −6.763856876661617, −5.774456096647490, −5.284209868984262, −4.213704008491916, −4.062606907724149, −3.341216908662718, −2.553345618439928, −1.705426992225661, −1.085506256948786, 0,
1.085506256948786, 1.705426992225661, 2.553345618439928, 3.341216908662718, 4.062606907724149, 4.213704008491916, 5.284209868984262, 5.774456096647490, 6.763856876661617, 7.565660146619165, 7.640797830546499, 8.374171937444584, 8.845864639293126, 9.220997802862314, 9.599264167652006, 10.34041118081979, 11.01277821271077, 11.42601761474547, 11.97930817091387, 12.90204161359496, 13.23760299352853, 13.76878611173295, 14.20763325188705, 14.74523383362685, 15.08459784126391