| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s − 4·11-s − 3·13-s + 14-s + 16-s + 17-s − 3·18-s − 4·22-s − 3·26-s + 28-s − 4·31-s + 32-s + 34-s − 3·36-s − 11·37-s − 10·41-s + 2·43-s − 4·44-s + 11·47-s + 49-s − 3·52-s − 53-s + 56-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s − 1.20·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.707·18-s − 0.852·22-s − 0.588·26-s + 0.188·28-s − 0.718·31-s + 0.176·32-s + 0.171·34-s − 1/2·36-s − 1.80·37-s − 1.56·41-s + 0.304·43-s − 0.603·44-s + 1.60·47-s + 1/7·49-s − 0.416·52-s − 0.137·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.41694960178630, −12.90761798039519, −12.41584494610138, −11.92101073882816, −11.74441492659283, −10.95699629749749, −10.63867016818272, −10.27658141388632, −9.687264519587259, −8.870056327274910, −8.721028952553093, −7.892921550505347, −7.679389217202584, −7.096116022313568, −6.587049609566845, −5.826819634761582, −5.503956052987362, −5.015029410903241, −4.738233944599589, −3.868547780323551, −3.289130199232387, −2.912447436525400, −2.109920154088987, −1.931420508859118, −0.7291894722792538, 0,
0.7291894722792538, 1.931420508859118, 2.109920154088987, 2.912447436525400, 3.289130199232387, 3.868547780323551, 4.738233944599589, 5.015029410903241, 5.503956052987362, 5.826819634761582, 6.587049609566845, 7.096116022313568, 7.679389217202584, 7.892921550505347, 8.721028952553093, 8.870056327274910, 9.687264519587259, 10.27658141388632, 10.63867016818272, 10.95699629749749, 11.74441492659283, 11.92101073882816, 12.41584494610138, 12.90761798039519, 13.41694960178630
