| L(s) = 1 | − 1.81·2-s − 3-s + 1.28·4-s + 3.62·5-s + 1.81·6-s + 1.49·7-s + 1.29·8-s + 9-s − 6.55·10-s − 1.96·11-s − 1.28·12-s + 4.09·13-s − 2.70·14-s − 3.62·15-s − 4.91·16-s − 4.14·17-s − 1.81·18-s + 5.10·19-s + 4.64·20-s − 1.49·21-s + 3.55·22-s − 7.47·23-s − 1.29·24-s + 8.10·25-s − 7.42·26-s − 27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.641·4-s + 1.61·5-s + 0.739·6-s + 0.564·7-s + 0.459·8-s + 0.333·9-s − 2.07·10-s − 0.591·11-s − 0.370·12-s + 1.13·13-s − 0.722·14-s − 0.934·15-s − 1.22·16-s − 1.00·17-s − 0.427·18-s + 1.17·19-s + 1.03·20-s − 0.325·21-s + 0.757·22-s − 1.55·23-s − 0.265·24-s + 1.62·25-s − 1.45·26-s − 0.192·27-s + 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 447 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8441122592\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8441122592\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 4.09T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 19 | \( 1 - 5.10T + 19T^{2} \) |
| 23 | \( 1 + 7.47T + 23T^{2} \) |
| 29 | \( 1 - 8.04T + 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 4.40T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 6.93T + 47T^{2} \) |
| 53 | \( 1 + 0.0232T + 53T^{2} \) |
| 59 | \( 1 + 5.36T + 59T^{2} \) |
| 61 | \( 1 - 3.76T + 61T^{2} \) |
| 67 | \( 1 + 2.10T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 1.72T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 5.30T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−10.76481639528149728746037440844, −10.13527562586055922696586611402, −9.411853085129375353358119872479, −8.571621748859271572587632547611, −7.62927966505692372150207634707, −6.41355653976592979149224844663, −5.66377333685181195755278906395, −4.51597561441106919004889638120, −2.28827330597645129559982451780, −1.18197635919208053076576947033,
1.18197635919208053076576947033, 2.28827330597645129559982451780, 4.51597561441106919004889638120, 5.66377333685181195755278906395, 6.41355653976592979149224844663, 7.62927966505692372150207634707, 8.571621748859271572587632547611, 9.411853085129375353358119872479, 10.13527562586055922696586611402, 10.76481639528149728746037440844
