L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 7-s − 8-s + 9-s − 10-s + 5·11-s − 12-s + 14-s − 15-s + 16-s + 17-s − 18-s − 19-s + 20-s + 21-s − 5·22-s + 24-s + 25-s − 27-s − 28-s − 2·29-s + 30-s − 32-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 + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s − 1.06·22-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−14.11307217039460, −13.76020359492912, −13.02063438189027, −12.62692246269587, −11.99412704229928, −11.71172248063555, −11.17095041543187, −10.66374137091748, −9.944578191014634, −9.858136874358277, −9.187162153441035, −8.714639298533925, −8.276961037476030, −7.444608731245985, −6.959858690384173, −6.466723176164044, −6.216902335381883, −5.421936616009443, −4.997362073648646, −4.109511428261507, −3.612966129184623, −2.991280777640606, −2.011402131065928, −1.579222070697351, −0.8709720119403842, 0,
0.8709720119403842, 1.579222070697351, 2.011402131065928, 2.991280777640606, 3.612966129184623, 4.109511428261507, 4.997362073648646, 5.421936616009443, 6.216902335381883, 6.466723176164044, 6.959858690384173, 7.444608731245985, 8.276961037476030, 8.714639298533925, 9.187162153441035, 9.858136874358277, 9.944578191014634, 10.66374137091748, 11.17095041543187, 11.71172248063555, 11.99412704229928, 12.62692246269587, 13.02063438189027, 13.76020359492912, 14.11307217039460