| L(s) = 1 | − 2.54·2-s + 0.532·3-s + 4.46·4-s − 3.72·5-s − 1.35·6-s − 4.87·7-s − 6.26·8-s − 2.71·9-s + 9.45·10-s + 4.55·11-s + 2.37·12-s + 12.3·14-s − 1.98·15-s + 6.99·16-s + 1.64·17-s + 6.90·18-s + 3.55·19-s − 16.6·20-s − 2.59·21-s − 11.5·22-s − 4.85·23-s − 3.33·24-s + 8.84·25-s − 3.04·27-s − 21.7·28-s + 0.682·29-s + 5.03·30-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.307·3-s + 2.23·4-s − 1.66·5-s − 0.552·6-s − 1.84·7-s − 2.21·8-s − 0.905·9-s + 2.99·10-s + 1.37·11-s + 0.685·12-s + 3.30·14-s − 0.511·15-s + 1.74·16-s + 0.400·17-s + 1.62·18-s + 0.816·19-s − 3.71·20-s − 0.565·21-s − 2.46·22-s − 1.01·23-s − 0.680·24-s + 1.76·25-s − 0.585·27-s − 4.10·28-s + 0.126·29-s + 0.919·30-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
| \(L(1)\) |
\(\approx\) |
\(0.09314453885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09314453885\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 - 0.532T + 3T^{2} \) |
| 5 | \( 1 + 3.72T + 5T^{2} \) |
| 7 | \( 1 + 4.87T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 - 0.682T + 29T^{2} \) |
| 37 | \( 1 + 8.45T + 37T^{2} \) |
| 41 | \( 1 - 1.77T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 9.80T + 47T^{2} \) |
| 53 | \( 1 - 1.80T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 + 0.394T + 67T^{2} \) |
| 71 | \( 1 + 1.02T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 6.69T + 83T^{2} \) |
| 89 | \( 1 + 3.60T + 89T^{2} \) |
| 97 | \( 1 + 3.84T + 97T^{2} \) |
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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
−8.402153752702016423871134692765, −7.66261328202449160253554722428, −6.95079936306093887777489638420, −6.58162556379281736546963125666, −5.69728508520833496773599888428, −4.08988043364058615057362546025, −3.29305352009984280234133791589, −3.00683675037291341356296056465, −1.48181613767363228678661334009, −0.21616450762799693886592418625,
0.21616450762799693886592418625, 1.48181613767363228678661334009, 3.00683675037291341356296056465, 3.29305352009984280234133791589, 4.08988043364058615057362546025, 5.69728508520833496773599888428, 6.58162556379281736546963125666, 6.95079936306093887777489638420, 7.66261328202449160253554722428, 8.402153752702016423871134692765
