L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 11-s + 2·13-s + 14-s + 16-s + 3·17-s − 7·19-s − 22-s + 3·23-s − 2·26-s − 28-s − 6·29-s − 4·31-s − 32-s − 3·34-s + 5·37-s + 7·38-s − 3·41-s − 4·43-s + 44-s − 3·46-s + 9·47-s − 6·49-s + 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.301·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 0.213·22-s + 0.625·23-s − 0.392·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.514·34-s + 0.821·37-s + 1.13·38-s − 0.468·41-s − 0.609·43-s + 0.150·44-s − 0.442·46-s + 1.31·47-s − 6/7·49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.920580997724999981772974865151, −7.29582787728866206975227086892, −6.43257434930820036861726787314, −6.00738490859124332731431189205, −5.01969653201601867452282357877, −3.97396849869356780971638398102, −3.28138172540388492063484699851, −2.22578169721846023348050680555, −1.29337664800075534098524334708, 0,
1.29337664800075534098524334708, 2.22578169721846023348050680555, 3.28138172540388492063484699851, 3.97396849869356780971638398102, 5.01969653201601867452282357877, 6.00738490859124332731431189205, 6.43257434930820036861726787314, 7.29582787728866206975227086892, 7.920580997724999981772974865151