| L(s) = 1 | − 2-s − 3.08·3-s + 4-s + 1.01·5-s + 3.08·6-s − 2.58·7-s − 8-s + 6.51·9-s − 1.01·10-s − 11-s − 3.08·12-s + 0.349·13-s + 2.58·14-s − 3.14·15-s + 16-s − 2.84·17-s − 6.51·18-s − 2.90·19-s + 1.01·20-s + 7.98·21-s + 22-s − 8.81·23-s + 3.08·24-s − 3.96·25-s − 0.349·26-s − 10.8·27-s − 2.58·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.78·3-s + 0.5·4-s + 0.455·5-s + 1.25·6-s − 0.978·7-s − 0.353·8-s + 2.17·9-s − 0.322·10-s − 0.301·11-s − 0.890·12-s + 0.0968·13-s + 0.691·14-s − 0.811·15-s + 0.250·16-s − 0.690·17-s − 1.53·18-s − 0.667·19-s + 0.227·20-s + 1.74·21-s + 0.213·22-s − 1.83·23-s + 0.629·24-s − 0.792·25-s − 0.0684·26-s − 2.08·27-s − 0.489·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2759709058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2759709058\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 5 | \( 1 - 1.01T + 5T^{2} \) |
| 7 | \( 1 + 2.58T + 7T^{2} \) |
| 13 | \( 1 - 0.349T + 13T^{2} \) |
| 17 | \( 1 + 2.84T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 + 8.81T + 23T^{2} \) |
| 29 | \( 1 - 6.00T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 - 3.52T + 41T^{2} \) |
| 43 | \( 1 - 0.166T + 43T^{2} \) |
| 47 | \( 1 + 1.10T + 47T^{2} \) |
| 53 | \( 1 + 4.21T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 + 9.09T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 2.45T + 79T^{2} \) |
| 83 | \( 1 + 8.13T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 - 4.63T + 97T^{2} \) |
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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
−8.338157371400589685723068984277, −7.52666630316484343277205687725, −6.64064183862800315330750367613, −6.09676191704615526582289913478, −5.93598498775655778440034877233, −4.73900447726255543562453354300, −4.04082438298451846253067182875, −2.65920715237384507331136653441, −1.61386763852511218445259956264, −0.35868728298879087745442457396,
0.35868728298879087745442457396, 1.61386763852511218445259956264, 2.65920715237384507331136653441, 4.04082438298451846253067182875, 4.73900447726255543562453354300, 5.93598498775655778440034877233, 6.09676191704615526582289913478, 6.64064183862800315330750367613, 7.52666630316484343277205687725, 8.338157371400589685723068984277
