| L(s) = 1 | − 3.04·3-s + 6.25·9-s + 3.20·11-s − 6.25·13-s + 1.78·17-s + 3.67·19-s − 3.21·23-s − 9.92·27-s + 2.88·29-s − 2.81·31-s − 9.73·33-s − 5.08·37-s + 19.0·39-s + 0.899·41-s + 7.26·43-s − 2.52·47-s − 5.42·51-s + 9.04·53-s − 11.1·57-s + 5.58·59-s − 10.9·61-s − 1.40·67-s + 9.78·69-s − 15.3·71-s − 7.01·73-s + 0.186·79-s + 11.4·81-s + ⋯ |
| L(s) = 1 | − 1.75·3-s + 2.08·9-s + 0.965·11-s − 1.73·13-s + 0.432·17-s + 0.843·19-s − 0.670·23-s − 1.90·27-s + 0.535·29-s − 0.506·31-s − 1.69·33-s − 0.835·37-s + 3.05·39-s + 0.140·41-s + 1.10·43-s − 0.367·47-s − 0.759·51-s + 1.24·53-s − 1.48·57-s + 0.727·59-s − 1.39·61-s − 0.171·67-s + 1.17·69-s − 1.82·71-s − 0.821·73-s + 0.0210·79-s + 1.26·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.04T + 3T^{2} \) |
| 11 | \( 1 - 3.20T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 - 1.78T + 17T^{2} \) |
| 19 | \( 1 - 3.67T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 + 5.08T + 37T^{2} \) |
| 41 | \( 1 - 0.899T + 41T^{2} \) |
| 43 | \( 1 - 7.26T + 43T^{2} \) |
| 47 | \( 1 + 2.52T + 47T^{2} \) |
| 53 | \( 1 - 9.04T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + 7.01T + 73T^{2} \) |
| 79 | \( 1 - 0.186T + 79T^{2} \) |
| 83 | \( 1 + 0.134T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 9.75T + 97T^{2} \) |
| 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
−7.25686570120070563903647266909, −6.61715146552927887377393203228, −5.81141404329230320631592425384, −5.44151059445852787949155755662, −4.64096043211700503766277519519, −4.16769847616037357393994449677, −3.07178157782085929849029226697, −1.89719172606298854138508513356, −0.981687766651249668491502880494, 0,
0.981687766651249668491502880494, 1.89719172606298854138508513356, 3.07178157782085929849029226697, 4.16769847616037357393994449677, 4.64096043211700503766277519519, 5.44151059445852787949155755662, 5.81141404329230320631592425384, 6.61715146552927887377393203228, 7.25686570120070563903647266909
