L(s) = 1 | + 2-s + 2.68·3-s + 4-s + 5-s + 2.68·6-s + 4.59·7-s + 8-s + 4.22·9-s + 10-s − 5.13·11-s + 2.68·12-s − 1.22·13-s + 4.59·14-s + 2.68·15-s + 16-s + 4.68·17-s + 4.22·18-s + 4.59·19-s + 20-s + 12.3·21-s − 5.13·22-s + 2.68·24-s + 25-s − 1.22·26-s + 3.28·27-s + 4.59·28-s + 3.37·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.55·3-s + 0.5·4-s + 0.447·5-s + 1.09·6-s + 1.73·7-s + 0.353·8-s + 1.40·9-s + 0.316·10-s − 1.54·11-s + 0.775·12-s − 0.338·13-s + 1.22·14-s + 0.693·15-s + 0.250·16-s + 1.13·17-s + 0.995·18-s + 1.05·19-s + 0.223·20-s + 2.69·21-s − 1.09·22-s + 0.548·24-s + 0.200·25-s − 0.239·26-s + 0.632·27-s + 0.868·28-s + 0.626·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.192384603\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.192384603\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 7 | \( 1 - 4.59T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 + 1.22T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 4.59T + 19T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 0.777T + 31T^{2} \) |
| 37 | \( 1 + 5.81T + 37T^{2} \) |
| 41 | \( 1 + 8.50T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 1.31T + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 - 4.88T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 + 8.93T + 89T^{2} \) |
| 97 | \( 1 - 18.0T + 97T^{2} \) |
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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
−8.239858054728734994516883463912, −7.55201525237360954871242933708, −7.13109420931190164676078627216, −5.69566366078123926162477055323, −5.10332184670167621500059390160, −4.66610523362160838657701776569, −3.44751216237722378240481449055, −2.92297660399449295998224733165, −2.06754207227145485152371487459, −1.43852526136901948256193569870,
1.43852526136901948256193569870, 2.06754207227145485152371487459, 2.92297660399449295998224733165, 3.44751216237722378240481449055, 4.66610523362160838657701776569, 5.10332184670167621500059390160, 5.69566366078123926162477055323, 7.13109420931190164676078627216, 7.55201525237360954871242933708, 8.239858054728734994516883463912