| L(s) = 1 | − 11.5·3-s + 89.6·7-s − 108.·9-s + 718.·11-s + 871.·13-s + 2.09e3·17-s + 476.·19-s − 1.03e3·21-s − 1.33e3·23-s + 4.07e3·27-s + 730.·29-s + 9.78e3·31-s − 8.33e3·33-s − 6.21e3·37-s − 1.01e4·39-s + 867.·41-s + 1.29e4·43-s − 2.61e4·47-s − 8.76e3·49-s − 2.42e4·51-s + 1.11e4·53-s − 5.52e3·57-s − 2.14e4·59-s + 3.26e4·61-s − 9.74e3·63-s − 3.72e4·67-s + 1.54e4·69-s + ⋯ |
| L(s) = 1 | − 0.743·3-s + 0.691·7-s − 0.447·9-s + 1.79·11-s + 1.43·13-s + 1.75·17-s + 0.303·19-s − 0.514·21-s − 0.524·23-s + 1.07·27-s + 0.161·29-s + 1.82·31-s − 1.33·33-s − 0.746·37-s − 1.06·39-s + 0.0806·41-s + 1.06·43-s − 1.72·47-s − 0.521·49-s − 1.30·51-s + 0.545·53-s − 0.225·57-s − 0.800·59-s + 1.12·61-s − 0.309·63-s − 1.01·67-s + 0.389·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.571446250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.571446250\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 11.5T + 243T^{2} \) |
| 7 | \( 1 - 89.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 718.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 871.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.09e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 476.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.33e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 730.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.21e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 867.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.29e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.26e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.83e4T + 8.58e9T^{2} \) |
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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
−9.549074370843198501016281698688, −8.568292660052433925727546986586, −7.959251056596268442470908402471, −6.61280647653204818628757578132, −6.07460559599376389567517415947, −5.20468325466481331712675456582, −4.09589385290479193563860499881, −3.19968280762892270639699099621, −1.44912720375443683576421989772, −0.887400802586041959305665358720,
0.887400802586041959305665358720, 1.44912720375443683576421989772, 3.19968280762892270639699099621, 4.09589385290479193563860499881, 5.20468325466481331712675456582, 6.07460559599376389567517415947, 6.61280647653204818628757578132, 7.959251056596268442470908402471, 8.568292660052433925727546986586, 9.549074370843198501016281698688
