L(s) = 1 | + 1.10·3-s + 5-s + 3.83·7-s − 1.78·9-s + 1.03·11-s + 4.66·13-s + 1.10·15-s + 0.694·17-s − 7.21·19-s + 4.22·21-s − 4.06·23-s + 25-s − 5.26·27-s + 6.12·29-s − 9.62·31-s + 1.14·33-s + 3.83·35-s + 8.08·37-s + 5.13·39-s + 1.83·41-s + 0.683·43-s − 1.78·45-s + 3.49·47-s + 7.73·49-s + 0.764·51-s + 4.11·53-s + 1.03·55-s + ⋯ |
L(s) = 1 | + 0.635·3-s + 0.447·5-s + 1.45·7-s − 0.596·9-s + 0.313·11-s + 1.29·13-s + 0.284·15-s + 0.168·17-s − 1.65·19-s + 0.921·21-s − 0.846·23-s + 0.200·25-s − 1.01·27-s + 1.13·29-s − 1.72·31-s + 0.198·33-s + 0.648·35-s + 1.32·37-s + 0.822·39-s + 0.286·41-s + 0.104·43-s − 0.266·45-s + 0.510·47-s + 1.10·49-s + 0.107·51-s + 0.564·53-s + 0.140·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.522519009\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.522519009\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 1.10T + 3T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 - 4.66T + 13T^{2} \) |
| 17 | \( 1 - 0.694T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 + 9.62T + 31T^{2} \) |
| 37 | \( 1 - 8.08T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 - 0.683T + 43T^{2} \) |
| 47 | \( 1 - 3.49T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 - 14.8T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 5.49T + 67T^{2} \) |
| 71 | \( 1 + 2.30T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 2.90T + 79T^{2} \) |
| 83 | \( 1 - 6.18T + 83T^{2} \) |
| 89 | \( 1 + 3.53T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−8.136329888274965657781206807180, −7.27210221333119127961089105156, −6.30302862474960446116280938512, −5.82877024077018669394600770565, −5.05238036318460202174851665066, −4.12122028192126509092797606443, −3.66724541016287189307367338424, −2.37381927136167625092852728953, −1.99763378734727015960775027795, −0.930679924436705612284529023499,
0.930679924436705612284529023499, 1.99763378734727015960775027795, 2.37381927136167625092852728953, 3.66724541016287189307367338424, 4.12122028192126509092797606443, 5.05238036318460202174851665066, 5.82877024077018669394600770565, 6.30302862474960446116280938512, 7.27210221333119127961089105156, 8.136329888274965657781206807180