| L(s) = 1 | + 4·7-s − 6·11-s − 2·13-s + 17-s − 4·19-s − 5·25-s + 4·31-s + 4·37-s − 6·41-s + 8·43-s + 9·49-s − 6·53-s + 4·61-s + 8·67-s + 2·73-s − 24·77-s − 8·79-s + 6·89-s − 8·91-s + 14·97-s + 18·101-s + 16·103-s + 6·107-s + 16·109-s + 6·113-s + 4·119-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.80·11-s − 0.554·13-s + 0.242·17-s − 0.917·19-s − 25-s + 0.718·31-s + 0.657·37-s − 0.937·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s + 0.512·61-s + 0.977·67-s + 0.234·73-s − 2.73·77-s − 0.900·79-s + 0.635·89-s − 0.838·91-s + 1.42·97-s + 1.79·101-s + 1.57·103-s + 0.580·107-s + 1.53·109-s + 0.564·113-s + 0.366·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835346867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835346867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−7.71319623971707540470742927921, −7.29484938101166108994999958784, −6.15777205658123834208045946258, −5.55324695398511660179242639164, −4.78136048868217964603743789674, −4.53177870298203396965614639892, −3.37176613840163753335045610201, −2.35027024394452387135343152508, −1.96477727170631410018279771361, −0.62289611347536456250264873990,
0.62289611347536456250264873990, 1.96477727170631410018279771361, 2.35027024394452387135343152508, 3.37176613840163753335045610201, 4.53177870298203396965614639892, 4.78136048868217964603743789674, 5.55324695398511660179242639164, 6.15777205658123834208045946258, 7.29484938101166108994999958784, 7.71319623971707540470742927921
