| L(s) = 1 | − 2-s − 4-s + 7-s + 3·8-s − 4·13-s − 14-s − 16-s − 2·17-s + 4·26-s − 28-s − 8·29-s − 4·31-s − 5·32-s + 2·34-s + 8·37-s − 4·41-s − 8·43-s − 12·47-s + 49-s + 4·52-s + 6·53-s + 3·56-s + 8·58-s + 8·59-s − 10·61-s + 4·62-s + 7·64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.377·7-s + 1.06·8-s − 1.10·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.784·26-s − 0.188·28-s − 1.48·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 1.31·37-s − 0.624·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s + 0.554·52-s + 0.824·53-s + 0.400·56-s + 1.05·58-s + 1.04·59-s − 1.28·61-s + 0.508·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.26834814879935, −13.19371959433883, −12.86277671333768, −12.04682724175667, −11.57120985825294, −11.27316214021724, −10.61882031431698, −10.18351208794394, −9.692743897063750, −9.422693548500675, −8.785435177820562, −8.459110411856713, −7.864651379135557, −7.444760054089469, −7.074434670351440, −6.455056624216699, −5.637422913467187, −5.315980007408387, −4.637249197879180, −4.376487727652538, −3.668936340755880, −3.049629895829808, −2.239843772554128, −1.750701505741390, −1.151289508193186, 0, 0,
1.151289508193186, 1.750701505741390, 2.239843772554128, 3.049629895829808, 3.668936340755880, 4.376487727652538, 4.637249197879180, 5.315980007408387, 5.637422913467187, 6.455056624216699, 7.074434670351440, 7.444760054089469, 7.864651379135557, 8.459110411856713, 8.785435177820562, 9.422693548500675, 9.692743897063750, 10.18351208794394, 10.61882031431698, 11.27316214021724, 11.57120985825294, 12.04682724175667, 12.86277671333768, 13.19371959433883, 13.26834814879935
