| L(s) = 1 | − i·3-s − 1.41i·5-s + 4.24·7-s − 9-s + i·11-s + 1.41i·13-s − 1.41·15-s + 4·17-s − 4.24i·21-s + 1.41·23-s + 2.99·25-s + i·27-s + 2.82i·29-s + 5.65·31-s + 33-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.632i·5-s + 1.60·7-s − 0.333·9-s + 0.301i·11-s + 0.392i·13-s − 0.365·15-s + 0.970·17-s − 0.925i·21-s + 0.294·23-s + 0.599·25-s + 0.192i·27-s + 0.525i·29-s + 1.01·31-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.277267633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.277267633\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 11 | \( 1 - iT \) |
| good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 - 1.41iT - 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.792076764003373348453057713232, −8.163778899645189207173615888244, −7.62830665197418615013024197031, −6.77123973065393617486861157355, −5.72372762801223228489484935276, −4.92827203803493560072649106339, −4.39488698699864898396544856450, −2.98703651236986330210254150709, −1.75668580941203280559863334319, −1.07318584335065884596681041491,
1.12702219151372418083213463366, 2.46372233463218496850274923491, 3.38040840936301923783496801082, 4.41823244727385393248963075392, 5.13693986210567929265349764102, 5.87271042287929020319073560524, 6.91757453408473470590056275405, 7.88508526511331582643514115458, 8.251264555526232627418508769317, 9.181174571751775612146485602598
