| L(s) = 1 | − 2.34·2-s + 3.51·4-s − 0.517·5-s − 3.56·8-s + 1.21·10-s + 3.73·11-s − 5.73·13-s + 1.33·16-s + 4.51·17-s + 19-s − 1.81·20-s − 8.76·22-s + 7.73·23-s − 4.73·25-s + 13.4·26-s + 6.18·29-s + 6.69·31-s + 3.98·32-s − 10.6·34-s + 9.46·37-s − 2.34·38-s + 1.84·40-s + 2.26·41-s + 1.30·43-s + 13.1·44-s − 18.1·46-s − 4.18·47-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.75·4-s − 0.231·5-s − 1.26·8-s + 0.384·10-s + 1.12·11-s − 1.58·13-s + 0.334·16-s + 1.09·17-s + 0.229·19-s − 0.406·20-s − 1.86·22-s + 1.61·23-s − 0.946·25-s + 2.64·26-s + 1.14·29-s + 1.20·31-s + 0.704·32-s − 1.81·34-s + 1.55·37-s − 0.381·38-s + 0.291·40-s + 0.354·41-s + 0.198·43-s + 1.97·44-s − 2.67·46-s − 0.609·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9254160149\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9254160149\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 + 0.517T + 5T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 5.73T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 23 | \( 1 - 7.73T + 23T^{2} \) |
| 29 | \( 1 - 6.18T + 29T^{2} \) |
| 31 | \( 1 - 6.69T + 31T^{2} \) |
| 37 | \( 1 - 9.46T + 37T^{2} \) |
| 41 | \( 1 - 2.26T + 41T^{2} \) |
| 43 | \( 1 - 1.30T + 43T^{2} \) |
| 47 | \( 1 + 4.18T + 47T^{2} \) |
| 53 | \( 1 + 13.6T + 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 + 0.947T + 71T^{2} \) |
| 73 | \( 1 + 3.03T + 73T^{2} \) |
| 79 | \( 1 - 6.06T + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 7.80T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.040503325192098615768478045077, −7.21816351703681730503245660755, −6.81853913916107782716655170822, −6.05079653621121375699049530090, −4.99732096875056796278920151951, −4.30829029838486827145038198902, −3.11432771621227519232839934232, −2.47217448521835483427554478505, −1.34656319873359377581141673665, −0.68050745680695219004773925274,
0.68050745680695219004773925274, 1.34656319873359377581141673665, 2.47217448521835483427554478505, 3.11432771621227519232839934232, 4.30829029838486827145038198902, 4.99732096875056796278920151951, 6.05079653621121375699049530090, 6.81853913916107782716655170822, 7.21816351703681730503245660755, 8.040503325192098615768478045077
