| L(s) = 1 | − 2-s + 4-s + 2.60·5-s − 4.35·7-s − 8-s − 2.60·10-s + 0.265·11-s + 4.05·13-s + 4.35·14-s + 16-s + 4.33·17-s + 4.48·19-s + 2.60·20-s − 0.265·22-s − 7.28·23-s + 1.77·25-s − 4.05·26-s − 4.35·28-s − 4.41·29-s − 2.50·31-s − 32-s − 4.33·34-s − 11.3·35-s − 1.01·37-s − 4.48·38-s − 2.60·40-s + 7.56·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.16·5-s − 1.64·7-s − 0.353·8-s − 0.822·10-s + 0.0799·11-s + 1.12·13-s + 1.16·14-s + 0.250·16-s + 1.05·17-s + 1.02·19-s + 0.581·20-s − 0.0565·22-s − 1.51·23-s + 0.354·25-s − 0.795·26-s − 0.822·28-s − 0.819·29-s − 0.450·31-s − 0.176·32-s − 0.742·34-s − 1.91·35-s − 0.166·37-s − 0.727·38-s − 0.411·40-s + 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.548381552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.548381552\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 - 0.265T + 11T^{2} \) |
| 13 | \( 1 - 4.05T + 13T^{2} \) |
| 17 | \( 1 - 4.33T + 17T^{2} \) |
| 19 | \( 1 - 4.48T + 19T^{2} \) |
| 23 | \( 1 + 7.28T + 23T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 2.50T + 31T^{2} \) |
| 37 | \( 1 + 1.01T + 37T^{2} \) |
| 41 | \( 1 - 7.56T + 41T^{2} \) |
| 43 | \( 1 - 8.16T + 43T^{2} \) |
| 47 | \( 1 + 4.04T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.90T + 59T^{2} \) |
| 61 | \( 1 + 4.59T + 61T^{2} \) |
| 67 | \( 1 + 2.97T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 + 4.42T + 73T^{2} \) |
| 79 | \( 1 - 5.32T + 79T^{2} \) |
| 83 | \( 1 + 14.1T + 83T^{2} \) |
| 89 | \( 1 + 2.33T + 89T^{2} \) |
| 97 | \( 1 - 2.03T + 97T^{2} \) |
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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
−7.76920504608322967165384962112, −7.18049641581143597514247808295, −6.34070243771781734757151935030, −5.78448339366457368206362760787, −5.63157131928348991297590153678, −3.98185357728617352276545213482, −3.39416182199311785046016574882, −2.58196028124413589061590538831, −1.66561161710134203124480908135, −0.69591577057176761881043051298,
0.69591577057176761881043051298, 1.66561161710134203124480908135, 2.58196028124413589061590538831, 3.39416182199311785046016574882, 3.98185357728617352276545213482, 5.63157131928348991297590153678, 5.78448339366457368206362760787, 6.34070243771781734757151935030, 7.18049641581143597514247808295, 7.76920504608322967165384962112
