| L(s) = 1 | − 2.41·2-s + 3-s + 3.84·4-s + 1.43·5-s − 2.41·6-s + 3.74·7-s − 4.46·8-s + 9-s − 3.47·10-s + 4.27·11-s + 3.84·12-s + 2.58·13-s − 9.04·14-s + 1.43·15-s + 3.11·16-s + 17-s − 2.41·18-s − 7.71·19-s + 5.52·20-s + 3.74·21-s − 10.3·22-s + 3.97·23-s − 4.46·24-s − 2.93·25-s − 6.26·26-s + 27-s + 14.3·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 0.577·3-s + 1.92·4-s + 0.642·5-s − 0.987·6-s + 1.41·7-s − 1.58·8-s + 0.333·9-s − 1.09·10-s + 1.28·11-s + 1.11·12-s + 0.718·13-s − 2.41·14-s + 0.371·15-s + 0.777·16-s + 0.242·17-s − 0.569·18-s − 1.76·19-s + 1.23·20-s + 0.816·21-s − 2.20·22-s + 0.828·23-s − 0.912·24-s − 0.586·25-s − 1.22·26-s + 0.192·27-s + 2.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.658826964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.658826964\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 - 1.43T + 5T^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 19 | \( 1 + 7.71T + 19T^{2} \) |
| 23 | \( 1 - 3.97T + 23T^{2} \) |
| 29 | \( 1 + 2.55T + 29T^{2} \) |
| 31 | \( 1 + 1.53T + 31T^{2} \) |
| 37 | \( 1 + 2.68T + 37T^{2} \) |
| 41 | \( 1 - 8.65T + 41T^{2} \) |
| 43 | \( 1 + 2.21T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 + 0.811T + 53T^{2} \) |
| 59 | \( 1 - 8.80T + 59T^{2} \) |
| 61 | \( 1 - 1.53T + 61T^{2} \) |
| 67 | \( 1 - 2.18T + 67T^{2} \) |
| 71 | \( 1 + 1.36T + 71T^{2} \) |
| 73 | \( 1 - 3.33T + 73T^{2} \) |
| 83 | \( 1 - 0.585T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 1.60T + 97T^{2} \) |
| 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
−8.611878732064763427601921228367, −8.016740513118055753863879294977, −7.25326753743597278011943407266, −6.54180594975180239016267479660, −5.79998805256832890888376298248, −4.57796813242387442708724149966, −3.72570696904172611441269523559, −2.27913519340406835992657473196, −1.77598952128462511039103848836, −1.01381305004215652564410974069,
1.01381305004215652564410974069, 1.77598952128462511039103848836, 2.27913519340406835992657473196, 3.72570696904172611441269523559, 4.57796813242387442708724149966, 5.79998805256832890888376298248, 6.54180594975180239016267479660, 7.25326753743597278011943407266, 8.016740513118055753863879294977, 8.611878732064763427601921228367
