| L(s) = 1 | − 0.302·3-s − 0.302·5-s − 2.90·9-s + 11-s + 5·13-s + 0.0916·15-s + 8.21·17-s + 6.90·19-s + 0.302·23-s − 4.90·25-s + 1.78·27-s + 6.69·29-s − 10.9·31-s − 0.302·33-s + 6.60·37-s − 1.51·39-s + 41-s − 4.39·43-s + 0.880·45-s + 12.6·47-s − 2.48·51-s − 6.30·53-s − 0.302·55-s − 2.09·57-s + 4.90·59-s − 4.21·61-s − 1.51·65-s + ⋯ |
| L(s) = 1 | − 0.174·3-s − 0.135·5-s − 0.969·9-s + 0.301·11-s + 1.38·13-s + 0.0236·15-s + 1.99·17-s + 1.58·19-s + 0.0631·23-s − 0.981·25-s + 0.344·27-s + 1.24·29-s − 1.95·31-s − 0.0527·33-s + 1.08·37-s − 0.242·39-s + 0.156·41-s − 0.670·43-s + 0.131·45-s + 1.83·47-s − 0.348·51-s − 0.865·53-s − 0.0408·55-s − 0.277·57-s + 0.639·59-s − 0.539·61-s − 0.187·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112482619\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112482619\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 0.302T + 3T^{2} \) |
| 5 | \( 1 + 0.302T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 8.21T + 17T^{2} \) |
| 19 | \( 1 - 6.90T + 19T^{2} \) |
| 23 | \( 1 - 0.302T + 23T^{2} \) |
| 29 | \( 1 - 6.69T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 - 6.60T + 37T^{2} \) |
| 43 | \( 1 + 4.39T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 + 6.30T + 53T^{2} \) |
| 59 | \( 1 - 4.90T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 - 2.21T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 9.39T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 9.51T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.71638990036421097168800748589, −7.36320937280697382074233550496, −6.20371412498216636535884655763, −5.74120433421869074987341278352, −5.31737732229422706927559206562, −4.16267759114975335419251961844, −3.38708661925172239682050548395, −2.96071605181874240499387731422, −1.55127218323864400537635803381, −0.78447760048948505192138535218,
0.78447760048948505192138535218, 1.55127218323864400537635803381, 2.96071605181874240499387731422, 3.38708661925172239682050548395, 4.16267759114975335419251961844, 5.31737732229422706927559206562, 5.74120433421869074987341278352, 6.20371412498216636535884655763, 7.36320937280697382074233550496, 7.71638990036421097168800748589
