| L(s) = 1 | + 2-s + 4-s − 0.701·5-s + 3.38·7-s + 8-s − 0.701·10-s − 3.84·11-s − 4.26·13-s + 3.38·14-s + 16-s + 6.75·17-s + 7.87·19-s − 0.701·20-s − 3.84·22-s − 1.63·23-s − 4.50·25-s − 4.26·26-s + 3.38·28-s + 2.14·29-s − 2.31·31-s + 32-s + 6.75·34-s − 2.37·35-s + 5.66·37-s + 7.87·38-s − 0.701·40-s − 1.47·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.313·5-s + 1.28·7-s + 0.353·8-s − 0.221·10-s − 1.15·11-s − 1.18·13-s + 0.905·14-s + 0.250·16-s + 1.63·17-s + 1.80·19-s − 0.156·20-s − 0.819·22-s − 0.341·23-s − 0.901·25-s − 0.836·26-s + 0.640·28-s + 0.398·29-s − 0.415·31-s + 0.176·32-s + 1.15·34-s − 0.401·35-s + 0.930·37-s + 1.27·38-s − 0.110·40-s − 0.229·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.276960215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.276960215\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 0.701T + 5T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 - 7.87T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 - 6.07T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 7.27T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 9.74T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 9.88T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 - 0.294T + 97T^{2} \) |
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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
−7.975192319470274082664848300195, −7.63966372849637862062075145065, −7.36054299656958216186110169244, −5.86192369330318863985249442360, −5.32933049269841674827301302852, −4.86481675408657615934761799888, −3.92297252288262429851749075216, −2.97185588556206463885115398913, −2.18318174101065620002665803264, −0.964306229897430743459721361927,
0.964306229897430743459721361927, 2.18318174101065620002665803264, 2.97185588556206463885115398913, 3.92297252288262429851749075216, 4.86481675408657615934761799888, 5.32933049269841674827301302852, 5.86192369330318863985249442360, 7.36054299656958216186110169244, 7.63966372849637862062075145065, 7.975192319470274082664848300195
