| L(s) = 1 | − 2.18·2-s − 0.879·3-s + 2.76·4-s + 3.84·5-s + 1.91·6-s + 1.42·7-s − 1.66·8-s − 2.22·9-s − 8.38·10-s − 0.518·11-s − 2.42·12-s − 13-s − 3.11·14-s − 3.37·15-s − 1.88·16-s + 8.19·17-s + 4.86·18-s + 6.55·19-s + 10.6·20-s − 1.25·21-s + 1.13·22-s − 9.07·23-s + 1.46·24-s + 9.74·25-s + 2.18·26-s + 4.59·27-s + 3.93·28-s + ⋯ |
| L(s) = 1 | − 1.54·2-s − 0.507·3-s + 1.38·4-s + 1.71·5-s + 0.783·6-s + 0.538·7-s − 0.589·8-s − 0.742·9-s − 2.65·10-s − 0.156·11-s − 0.701·12-s − 0.277·13-s − 0.831·14-s − 0.871·15-s − 0.472·16-s + 1.98·17-s + 1.14·18-s + 1.50·19-s + 2.37·20-s − 0.273·21-s + 0.241·22-s − 1.89·23-s + 0.299·24-s + 1.94·25-s + 0.428·26-s + 0.884·27-s + 0.744·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9410930351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9410930351\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 3 | \( 1 + 0.879T + 3T^{2} \) |
| 5 | \( 1 - 3.84T + 5T^{2} \) |
| 7 | \( 1 - 1.42T + 7T^{2} \) |
| 11 | \( 1 + 0.518T + 11T^{2} \) |
| 17 | \( 1 - 8.19T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 + 9.07T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 + 0.491T + 47T^{2} \) |
| 53 | \( 1 + 5.23T + 53T^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 + 2.46T + 61T^{2} \) |
| 67 | \( 1 + 0.907T + 67T^{2} \) |
| 71 | \( 1 - 2.88T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 5.21T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−9.727962209863298242128824178633, −9.076510554907443944803413276356, −7.930659700186435860026657731670, −7.60582315498483359558395832182, −6.16407575337135041335070220224, −5.80089799058242886708039883829, −4.91950270849268831943478938635, −2.96719203803021306537801322748, −1.90423592347971353930149542203, −0.965582729133367359644980774068,
0.965582729133367359644980774068, 1.90423592347971353930149542203, 2.96719203803021306537801322748, 4.91950270849268831943478938635, 5.80089799058242886708039883829, 6.16407575337135041335070220224, 7.60582315498483359558395832182, 7.930659700186435860026657731670, 9.076510554907443944803413276356, 9.727962209863298242128824178633
