L(s) = 1 | + 1.39·2-s + 3-s − 0.0671·4-s − 5-s + 1.39·6-s − 1.27·7-s − 2.87·8-s + 9-s − 1.39·10-s − 0.0671·12-s + 1.41·13-s − 1.77·14-s − 15-s − 3.86·16-s − 5·17-s + 1.39·18-s − 0.158·19-s + 0.0671·20-s − 1.27·21-s − 5.00·23-s − 2.87·24-s + 25-s + 1.97·26-s + 27-s + 0.0858·28-s − 6.27·29-s − 1.39·30-s + ⋯ |
L(s) = 1 | + 0.983·2-s + 0.577·3-s − 0.0335·4-s − 0.447·5-s + 0.567·6-s − 0.482·7-s − 1.01·8-s + 0.333·9-s − 0.439·10-s − 0.0193·12-s + 0.393·13-s − 0.474·14-s − 0.258·15-s − 0.965·16-s − 1.21·17-s + 0.327·18-s − 0.0362·19-s + 0.0150·20-s − 0.278·21-s − 1.04·23-s − 0.586·24-s + 0.200·25-s + 0.386·26-s + 0.192·27-s + 0.0162·28-s − 1.16·29-s − 0.253·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 1.39T + 2T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + 0.158T + 19T^{2} \) |
| 23 | \( 1 + 5.00T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 4.69T + 37T^{2} \) |
| 41 | \( 1 + 7.58T + 41T^{2} \) |
| 43 | \( 1 + 5.41T + 43T^{2} \) |
| 47 | \( 1 + 8.23T + 47T^{2} \) |
| 53 | \( 1 - 9.36T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 6.77T + 71T^{2} \) |
| 73 | \( 1 + 8.66T + 73T^{2} \) |
| 79 | \( 1 - 2.54T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 6.41T + 97T^{2} \) |
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
−8.691671549643162166415160121665, −8.317460341874348779697389768248, −7.08139534321304596693048304990, −6.44098619685757026685400425280, −5.50063552856583888126251069102, −4.54959977968657560593957244979, −3.82381326750442774592669024989, −3.19384974760876895743772796387, −2.03613793703779452970580103067, 0,
2.03613793703779452970580103067, 3.19384974760876895743772796387, 3.82381326750442774592669024989, 4.54959977968657560593957244979, 5.50063552856583888126251069102, 6.44098619685757026685400425280, 7.08139534321304596693048304990, 8.317460341874348779697389768248, 8.691671549643162166415160121665