| L(s) = 1 | − 3-s + 9-s − 2·13-s + 6·17-s + 4·19-s − 4·23-s − 27-s + 6·29-s − 6·37-s + 2·39-s − 6·41-s + 4·43-s + 8·47-s − 6·51-s − 14·53-s − 4·57-s + 4·59-s + 2·61-s − 12·67-s + 4·69-s − 12·71-s + 10·73-s + 8·79-s + 81-s − 4·83-s − 6·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.834·23-s − 0.192·27-s + 1.11·29-s − 0.986·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.840·51-s − 1.92·53-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 1.46·67-s + 0.481·69-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s − 0.643·87-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.53691827628255, −14.90060296247758, −14.26692499910846, −13.91049890460498, −13.40022636562135, −12.41773501894161, −12.26397065301538, −11.87831114137192, −11.18526428474996, −10.47399876469071, −10.13272352518594, −9.595770884563994, −9.030155551116297, −8.099716941489281, −7.834787237600402, −7.080747070324360, −6.621932901595435, −5.748269805750800, −5.463240820966864, −4.776022635523418, −4.106539204676600, −3.313096513991061, −2.745094166675677, −1.697433735765820, −1.025000530953021, 0,
1.025000530953021, 1.697433735765820, 2.745094166675677, 3.313096513991061, 4.106539204676600, 4.776022635523418, 5.463240820966864, 5.748269805750800, 6.621932901595435, 7.080747070324360, 7.834787237600402, 8.099716941489281, 9.030155551116297, 9.595770884563994, 10.13272352518594, 10.47399876469071, 11.18526428474996, 11.87831114137192, 12.26397065301538, 12.41773501894161, 13.40022636562135, 13.91049890460498, 14.26692499910846, 14.90060296247758, 15.53691827628255
