| L(s) = 1 | + 1.59·2-s + 2.05·3-s + 0.557·4-s − 4.20·5-s + 3.29·6-s + 2.52·7-s − 2.30·8-s + 1.24·9-s − 6.72·10-s − 4.89·11-s + 1.14·12-s + 4.03·14-s − 8.66·15-s − 4.80·16-s + 6.79·17-s + 1.98·18-s + 1.45·19-s − 2.34·20-s + 5.19·21-s − 7.83·22-s + 8.11·23-s − 4.75·24-s + 12.6·25-s − 3.61·27-s + 1.40·28-s + 0.475·29-s − 13.8·30-s + ⋯ |
| L(s) = 1 | + 1.13·2-s + 1.18·3-s + 0.278·4-s − 1.88·5-s + 1.34·6-s + 0.952·7-s − 0.815·8-s + 0.414·9-s − 2.12·10-s − 1.47·11-s + 0.331·12-s + 1.07·14-s − 2.23·15-s − 1.20·16-s + 1.64·17-s + 0.468·18-s + 0.333·19-s − 0.524·20-s + 1.13·21-s − 1.67·22-s + 1.69·23-s − 0.970·24-s + 2.53·25-s − 0.696·27-s + 0.265·28-s + 0.0883·29-s − 2.52·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5239 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.382063703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.382063703\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 3 | \( 1 - 2.05T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 23 | \( 1 - 8.11T + 23T^{2} \) |
| 29 | \( 1 - 0.475T + 29T^{2} \) |
| 37 | \( 1 + 3.66T + 37T^{2} \) |
| 41 | \( 1 + 1.38T + 41T^{2} \) |
| 43 | \( 1 - 8.71T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 5.10T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 1.57T + 73T^{2} \) |
| 79 | \( 1 - 0.974T + 79T^{2} \) |
| 83 | \( 1 + 8.20T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 3.24T + 97T^{2} \) |
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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
−8.143867856439421915074244464909, −7.55983598633779075036391041658, −7.14628209127532364708027238085, −5.61003435395149854177028717281, −5.06477098066662583755452042362, −4.44797166594285263926810870243, −3.57086421225168830990763526620, −3.17960793412284941979818326555, −2.49927747023785221472243059220, −0.78440492446901720598716222563,
0.78440492446901720598716222563, 2.49927747023785221472243059220, 3.17960793412284941979818326555, 3.57086421225168830990763526620, 4.44797166594285263926810870243, 5.06477098066662583755452042362, 5.61003435395149854177028717281, 7.14628209127532364708027238085, 7.55983598633779075036391041658, 8.143867856439421915074244464909
