| L(s) = 1 | + 1.42·3-s − 2.11·5-s − 3.55·7-s − 0.970·9-s − 4.05·13-s − 3.00·15-s − 4.39·17-s + 19-s − 5.06·21-s − 8.50·23-s − 0.545·25-s − 5.65·27-s + 8.32·29-s − 0.583·31-s + 7.51·35-s − 2.92·37-s − 5.77·39-s + 0.892·41-s + 8.63·43-s + 2.04·45-s − 0.756·47-s + 5.66·49-s − 6.26·51-s + 1.53·53-s + 1.42·57-s − 13.9·59-s − 14.9·61-s + ⋯ |
| L(s) = 1 | + 0.822·3-s − 0.943·5-s − 1.34·7-s − 0.323·9-s − 1.12·13-s − 0.776·15-s − 1.06·17-s + 0.229·19-s − 1.10·21-s − 1.77·23-s − 0.109·25-s − 1.08·27-s + 1.54·29-s − 0.104·31-s + 1.26·35-s − 0.481·37-s − 0.925·39-s + 0.139·41-s + 1.31·43-s + 0.305·45-s − 0.110·47-s + 0.809·49-s − 0.877·51-s + 0.210·53-s + 0.188·57-s − 1.81·59-s − 1.90·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4669259342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4669259342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.42T + 3T^{2} \) |
| 5 | \( 1 + 2.11T + 5T^{2} \) |
| 7 | \( 1 + 3.55T + 7T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 23 | \( 1 + 8.50T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 + 0.583T + 31T^{2} \) |
| 37 | \( 1 + 2.92T + 37T^{2} \) |
| 41 | \( 1 - 0.892T + 41T^{2} \) |
| 43 | \( 1 - 8.63T + 43T^{2} \) |
| 47 | \( 1 + 0.756T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 + 14.9T + 61T^{2} \) |
| 67 | \( 1 - 1.09T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 - 5.34T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
| 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
−7.77106535362319442281670712309, −7.21276934558401348935293958676, −6.37114312451804468566755309732, −5.87702222185157990655507363733, −4.67347318226854759359781876396, −4.12286481131765205172026396811, −3.32387809400301313167032625188, −2.78745722260548417018828984855, −2.03987390820856151826422156519, −0.28857198269905006721847376186,
0.28857198269905006721847376186, 2.03987390820856151826422156519, 2.78745722260548417018828984855, 3.32387809400301313167032625188, 4.12286481131765205172026396811, 4.67347318226854759359781876396, 5.87702222185157990655507363733, 6.37114312451804468566755309732, 7.21276934558401348935293958676, 7.77106535362319442281670712309
