L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 11-s − 3.46·13-s − 14-s + 16-s + 3.46·17-s − 20-s − 22-s + 1.46·23-s + 25-s − 3.46·26-s − 28-s − 8.92·29-s + 10.9·31-s + 32-s + 3.46·34-s + 35-s + 3.46·37-s − 40-s − 2·41-s − 44-s + 1.46·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.960·13-s − 0.267·14-s + 0.250·16-s + 0.840·17-s − 0.223·20-s − 0.213·22-s + 0.305·23-s + 0.200·25-s − 0.679·26-s − 0.188·28-s − 1.65·29-s + 1.96·31-s + 0.176·32-s + 0.594·34-s + 0.169·35-s + 0.569·37-s − 0.158·40-s − 0.312·41-s − 0.150·44-s + 0.215·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 3.46T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 + 1.07T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 8.53T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.67748903489183673708373646158, −6.83037374498026632908238186793, −6.18747179445599159935963859434, −5.36170187812529085077880780259, −4.77002518917864082764774613405, −4.00815799229655652624240862658, −3.15013242564107021027956354890, −2.60092616688986602321141820903, −1.39740166594518336368911085882, 0,
1.39740166594518336368911085882, 2.60092616688986602321141820903, 3.15013242564107021027956354890, 4.00815799229655652624240862658, 4.77002518917864082764774613405, 5.36170187812529085077880780259, 6.18747179445599159935963859434, 6.83037374498026632908238186793, 7.67748903489183673708373646158