| L(s) = 1 | − 2-s + 2.65·3-s + 4-s − 5-s − 2.65·6-s + 4.81·7-s − 8-s + 4.05·9-s + 10-s − 2.04·11-s + 2.65·12-s + 4.43·13-s − 4.81·14-s − 2.65·15-s + 16-s − 0.381·17-s − 4.05·18-s − 2.89·19-s − 20-s + 12.7·21-s + 2.04·22-s − 4.16·23-s − 2.65·24-s + 25-s − 4.43·26-s + 2.81·27-s + 4.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.53·3-s + 0.5·4-s − 0.447·5-s − 1.08·6-s + 1.81·7-s − 0.353·8-s + 1.35·9-s + 0.316·10-s − 0.615·11-s + 0.766·12-s + 1.22·13-s − 1.28·14-s − 0.685·15-s + 0.250·16-s − 0.0925·17-s − 0.956·18-s − 0.663·19-s − 0.223·20-s + 2.79·21-s + 0.435·22-s − 0.869·23-s − 0.542·24-s + 0.200·25-s − 0.869·26-s + 0.540·27-s + 0.909·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.918692392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.918692392\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 7 | \( 1 - 4.81T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 4.43T + 13T^{2} \) |
| 17 | \( 1 + 0.381T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 0.317T + 29T^{2} \) |
| 31 | \( 1 - 4.56T + 31T^{2} \) |
| 37 | \( 1 - 5.44T + 37T^{2} \) |
| 41 | \( 1 - 0.300T + 41T^{2} \) |
| 43 | \( 1 - 6.07T + 43T^{2} \) |
| 47 | \( 1 - 0.0416T + 47T^{2} \) |
| 53 | \( 1 - 3.14T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 + 3.88T + 61T^{2} \) |
| 67 | \( 1 - 0.170T + 67T^{2} \) |
| 71 | \( 1 - 4.22T + 71T^{2} \) |
| 73 | \( 1 - 3.15T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 2.66T + 83T^{2} \) |
| 89 | \( 1 + 2.34T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.395539931166642770243719079412, −7.945281537131182311002463148579, −7.57352888602339153524453926175, −6.48748148404482746666603770982, −5.44575787763249155513398372911, −4.36166244683333081748836379461, −3.82967788738423705779140133826, −2.65726586917636423342374655940, −2.02852534609416987573573257428, −1.08923076560461285086107966430,
1.08923076560461285086107966430, 2.02852534609416987573573257428, 2.65726586917636423342374655940, 3.82967788738423705779140133826, 4.36166244683333081748836379461, 5.44575787763249155513398372911, 6.48748148404482746666603770982, 7.57352888602339153524453926175, 7.945281537131182311002463148579, 8.395539931166642770243719079412
