| L(s) = 1 | − 2.18·3-s + 3.70·7-s + 1.76·9-s + 3.18·11-s + 1.94·13-s − 4.66·17-s − 19-s − 8.08·21-s + 5.23·23-s + 2.70·27-s − 1.66·29-s + 3.46·31-s − 6.94·33-s + 7.18·37-s − 4.23·39-s − 0.717·41-s − 1.36·43-s + 5.37·47-s + 6.71·49-s + 10.1·51-s − 0.0435·53-s + 2.18·57-s + 7.88·59-s − 4.98·61-s + 6.52·63-s + 14.5·67-s − 11.4·69-s + ⋯ |
| L(s) = 1 | − 1.25·3-s + 1.39·7-s + 0.586·9-s + 0.959·11-s + 0.538·13-s − 1.13·17-s − 0.229·19-s − 1.76·21-s + 1.09·23-s + 0.520·27-s − 0.308·29-s + 0.622·31-s − 1.20·33-s + 1.18·37-s − 0.678·39-s − 0.112·41-s − 0.207·43-s + 0.784·47-s + 0.959·49-s + 1.42·51-s − 0.00597·53-s + 0.289·57-s + 1.02·59-s − 0.638·61-s + 0.821·63-s + 1.77·67-s − 1.37·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680284725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680284725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 + 0.717T + 41T^{2} \) |
| 43 | \( 1 + 1.36T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 + 0.0435T + 53T^{2} \) |
| 59 | \( 1 - 7.88T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 + 4.88T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 9.76T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.80769913494492638857657193343, −7.01554040738511969150342405872, −6.39766001129288060830774920886, −5.83854201817177684553759768572, −4.97769824209274020016978723008, −4.57391262225006853056484560173, −3.82801562942688876995209570554, −2.52844829858437837560860272850, −1.50508772012040152172092596321, −0.75951365402736990949085760025,
0.75951365402736990949085760025, 1.50508772012040152172092596321, 2.52844829858437837560860272850, 3.82801562942688876995209570554, 4.57391262225006853056484560173, 4.97769824209274020016978723008, 5.83854201817177684553759768572, 6.39766001129288060830774920886, 7.01554040738511969150342405872, 7.80769913494492638857657193343
