L(s) = 1 | − 3-s + 5-s + 3.83·7-s + 9-s − 1.14·11-s − 3.53·13-s − 15-s − 6.97·17-s − 19-s − 3.83·21-s − 8.97·23-s + 25-s − 27-s + 0.853·29-s − 4.39·31-s + 1.14·33-s + 3.83·35-s + 1.83·37-s + 3.53·39-s − 8.51·41-s − 10.1·43-s + 45-s + 0.978·47-s + 7.68·49-s + 6.97·51-s + 6.97·53-s − 1.14·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.44·7-s + 0.333·9-s − 0.345·11-s − 0.981·13-s − 0.258·15-s − 1.69·17-s − 0.229·19-s − 0.836·21-s − 1.87·23-s + 0.200·25-s − 0.192·27-s + 0.158·29-s − 0.789·31-s + 0.199·33-s + 0.647·35-s + 0.301·37-s + 0.566·39-s − 1.33·41-s − 1.54·43-s + 0.149·45-s + 0.142·47-s + 1.09·49-s + 0.977·51-s + 0.958·53-s − 0.154·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 1.14T + 11T^{2} \) |
| 13 | \( 1 + 3.53T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 23 | \( 1 + 8.97T + 23T^{2} \) |
| 29 | \( 1 - 0.853T + 29T^{2} \) |
| 31 | \( 1 + 4.39T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 + 8.51T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 0.978T + 47T^{2} \) |
| 53 | \( 1 - 6.97T + 53T^{2} \) |
| 59 | \( 1 + 6.29T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 - 0.585T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.37T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 - 1.24T + 97T^{2} \) |
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
−8.517766318783849100454951857735, −7.919599124964349840411590387707, −7.03361229846020653964615577410, −6.27722403447369106013136814590, −5.28173602651481189475307351964, −4.81189396437695287288749279242, −3.98507818675218835996479153465, −2.31469844919040378437852117515, −1.76583808066119081406025429207, 0,
1.76583808066119081406025429207, 2.31469844919040378437852117515, 3.98507818675218835996479153465, 4.81189396437695287288749279242, 5.28173602651481189475307351964, 6.27722403447369106013136814590, 7.03361229846020653964615577410, 7.919599124964349840411590387707, 8.517766318783849100454951857735