| L(s) = 1 | + 1.53·2-s + 0.347·4-s + 2.53·5-s + 0.532·7-s − 2.53·8-s + 3.87·10-s + 5.10·11-s + 4.06·13-s + 0.815·14-s − 4.57·16-s − 1.94·17-s + 0.879·20-s + 7.82·22-s − 3.04·23-s + 1.41·25-s + 6.22·26-s + 0.184·28-s − 1.61·29-s + 9.87·31-s − 1.94·32-s − 2.97·34-s + 1.34·35-s + 6.10·37-s − 6.41·40-s + 8.47·41-s − 0.177·43-s + 1.77·44-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.173·4-s + 1.13·5-s + 0.201·7-s − 0.895·8-s + 1.22·10-s + 1.53·11-s + 1.12·13-s + 0.217·14-s − 1.14·16-s − 0.471·17-s + 0.196·20-s + 1.66·22-s − 0.634·23-s + 0.282·25-s + 1.22·26-s + 0.0349·28-s − 0.299·29-s + 1.77·31-s − 0.343·32-s − 0.510·34-s + 0.227·35-s + 1.00·37-s − 1.01·40-s + 1.32·41-s − 0.0270·43-s + 0.267·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.200954923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.200954923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 11 | \( 1 - 5.10T + 11T^{2} \) |
| 13 | \( 1 - 4.06T + 13T^{2} \) |
| 17 | \( 1 + 1.94T + 17T^{2} \) |
| 23 | \( 1 + 3.04T + 23T^{2} \) |
| 29 | \( 1 + 1.61T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 37 | \( 1 - 6.10T + 37T^{2} \) |
| 41 | \( 1 - 8.47T + 41T^{2} \) |
| 43 | \( 1 + 0.177T + 43T^{2} \) |
| 47 | \( 1 + 7.55T + 47T^{2} \) |
| 53 | \( 1 - 9.90T + 53T^{2} \) |
| 59 | \( 1 - 3.81T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 - 5.92T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.849666990218306284426578389868, −7.953449545615640000100835189648, −6.60042384093963534853645336617, −6.21637233083310340510105185052, −5.77283443140701463812373851972, −4.65464582139483557256417580821, −4.12961837170222476146029787514, −3.24953907009413870966778340982, −2.18684791128046091260184426860, −1.14984409452821762277715860013,
1.14984409452821762277715860013, 2.18684791128046091260184426860, 3.24953907009413870966778340982, 4.12961837170222476146029787514, 4.65464582139483557256417580821, 5.77283443140701463812373851972, 6.21637233083310340510105185052, 6.60042384093963534853645336617, 7.953449545615640000100835189648, 8.849666990218306284426578389868
