| L(s) = 1 | + 1.04·2-s − 2.73·3-s − 0.902·4-s − 5-s − 2.86·6-s + 7-s − 3.04·8-s + 4.46·9-s − 1.04·10-s + 2.46·12-s + 4.59·13-s + 1.04·14-s + 2.73·15-s − 1.38·16-s − 1.64·17-s + 4.67·18-s − 2.01·19-s + 0.902·20-s − 2.73·21-s − 5.80·23-s + 8.30·24-s + 25-s + 4.81·26-s − 4.00·27-s − 0.902·28-s + 4.38·29-s + 2.86·30-s + ⋯ |
| L(s) = 1 | + 0.740·2-s − 1.57·3-s − 0.451·4-s − 0.447·5-s − 1.16·6-s + 0.377·7-s − 1.07·8-s + 1.48·9-s − 0.331·10-s + 0.711·12-s + 1.27·13-s + 0.280·14-s + 0.705·15-s − 0.345·16-s − 0.398·17-s + 1.10·18-s − 0.461·19-s + 0.201·20-s − 0.596·21-s − 1.21·23-s + 1.69·24-s + 0.200·25-s + 0.944·26-s − 0.770·27-s − 0.170·28-s + 0.813·29-s + 0.522·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7704295876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7704295876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 + 1.64T + 17T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 5.80T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 9.12T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 + 7.18T + 43T^{2} \) |
| 47 | \( 1 - 5.94T + 47T^{2} \) |
| 53 | \( 1 + 1.11T + 53T^{2} \) |
| 59 | \( 1 + 0.538T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 - 0.579T + 71T^{2} \) |
| 73 | \( 1 - 2.20T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 - 18.6T + 89T^{2} \) |
| 97 | \( 1 - 19.1T + 97T^{2} \) |
| 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
−8.505470576880445529782681709396, −7.50848751063161393110432722078, −6.41083074993517191426584450417, −6.21457500646517330366767448338, −5.28791760244916717157621295804, −4.80737812881654202250637489916, −4.03349577436447515989378443929, −3.40345152019431069362795682641, −1.76370961733339695974179418142, −0.48620824723451976600370813515,
0.48620824723451976600370813515, 1.76370961733339695974179418142, 3.40345152019431069362795682641, 4.03349577436447515989378443929, 4.80737812881654202250637489916, 5.28791760244916717157621295804, 6.21457500646517330366767448338, 6.41083074993517191426584450417, 7.50848751063161393110432722078, 8.505470576880445529782681709396
