| L(s) = 1 | + 2.56·3-s + 5-s + 4.56·7-s + 3.56·9-s + 4·11-s − 2.56·13-s + 2.56·15-s − 0.561·17-s + 19-s + 11.6·21-s − 5.68·23-s + 25-s + 1.43·27-s + 3.43·29-s − 5.12·31-s + 10.2·33-s + 4.56·35-s + 1.12·37-s − 6.56·39-s + 8.24·41-s + 2·43-s + 3.56·45-s + 3.12·47-s + 13.8·49-s − 1.43·51-s − 6.56·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s + 1.72·7-s + 1.18·9-s + 1.20·11-s − 0.710·13-s + 0.661·15-s − 0.136·17-s + 0.229·19-s + 2.54·21-s − 1.18·23-s + 0.200·25-s + 0.276·27-s + 0.638·29-s − 0.920·31-s + 1.78·33-s + 0.771·35-s + 0.184·37-s − 1.05·39-s + 1.28·41-s + 0.304·43-s + 0.530·45-s + 0.455·47-s + 1.97·49-s − 0.201·51-s − 0.901·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.076850472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.076850472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 4.56T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 - 8.24T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 6.56T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 - 4.87T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 - 14T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−7.969651075274051660305262672216, −7.73481058444903123996088386283, −6.86122411739726078262032028020, −5.93249128047939885354153600072, −5.02318358195042063056795851237, −4.29270344077333594334145137561, −3.69618932089459072350032122476, −2.54399362690727469357695014806, −1.98432254075896860655833417497, −1.24543955199195850448238606316,
1.24543955199195850448238606316, 1.98432254075896860655833417497, 2.54399362690727469357695014806, 3.69618932089459072350032122476, 4.29270344077333594334145137561, 5.02318358195042063056795851237, 5.93249128047939885354153600072, 6.86122411739726078262032028020, 7.73481058444903123996088386283, 7.969651075274051660305262672216
