| L(s) = 1 | + 5.56·2-s + 3·3-s + 23.0·4-s + 16.7·6-s − 7·7-s + 83.5·8-s + 9·9-s + 23.2·11-s + 69.0·12-s − 46.5·13-s − 38.9·14-s + 281.·16-s + 76.0·17-s + 50.1·18-s − 114.·19-s − 21·21-s + 129.·22-s + 113.·23-s + 250.·24-s − 259.·26-s + 27·27-s − 161.·28-s + 120.·29-s − 182.·31-s + 897.·32-s + 69.8·33-s + 423.·34-s + ⋯ |
| L(s) = 1 | + 1.96·2-s + 0.577·3-s + 2.87·4-s + 1.13·6-s − 0.377·7-s + 3.69·8-s + 0.333·9-s + 0.638·11-s + 1.66·12-s − 0.992·13-s − 0.744·14-s + 4.39·16-s + 1.08·17-s + 0.656·18-s − 1.38·19-s − 0.218·21-s + 1.25·22-s + 1.03·23-s + 2.13·24-s − 1.95·26-s + 0.192·27-s − 1.08·28-s + 0.771·29-s − 1.05·31-s + 4.95·32-s + 0.368·33-s + 2.13·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.624192081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.624192081\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 5.56T + 8T^{2} \) |
| 11 | \( 1 - 23.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 182.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 322.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 93.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 452.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 402.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 495.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 265.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 594.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 487.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 21.5T + 9.12e5T^{2} \) |
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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
−10.67677711290748991068956150861, −9.871167172322302991149943626082, −8.452277425099117969931589980028, −7.21020933624159381362399771471, −6.72055453061693550594764804643, −5.56806456497069632191265962360, −4.65955201937060897914681147556, −3.67339395164243619753417093416, −2.86246781913768291286104228164, −1.71087048605369989933938876335,
1.71087048605369989933938876335, 2.86246781913768291286104228164, 3.67339395164243619753417093416, 4.65955201937060897914681147556, 5.56806456497069632191265962360, 6.72055453061693550594764804643, 7.21020933624159381362399771471, 8.452277425099117969931589980028, 9.871167172322302991149943626082, 10.67677711290748991068956150861
