| L(s) = 1 | − 4.24·5-s − 7.07·13-s + 2·17-s + 12.9·25-s − 9.89·29-s − 9.89·37-s − 10·41-s − 7·49-s − 7.07·53-s + 15.5·61-s + 30·65-s − 6·73-s − 8.48·85-s − 16·89-s + 8·97-s − 15.5·101-s + 18.3·109-s − 16·113-s + ⋯ |
| L(s) = 1 | − 1.89·5-s − 1.96·13-s + 0.485·17-s + 2.59·25-s − 1.83·29-s − 1.62·37-s − 1.56·41-s − 49-s − 0.971·53-s + 1.99·61-s + 3.72·65-s − 0.702·73-s − 0.920·85-s − 1.69·89-s + 0.812·97-s − 1.54·101-s + 1.76·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2811959114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2811959114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.24T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.89T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.66401402942644815708678744089, −7.17303914609409998231984775398, −6.70892675767121809281826838397, −5.32032384440731540182737062962, −5.00383105898729906472075176605, −4.12490766293577804674752998992, −3.52478201670782748594783877058, −2.83594339387032586226223705959, −1.71221558183860800261107532257, −0.24643323815737964398035768586,
0.24643323815737964398035768586, 1.71221558183860800261107532257, 2.83594339387032586226223705959, 3.52478201670782748594783877058, 4.12490766293577804674752998992, 5.00383105898729906472075176605, 5.32032384440731540182737062962, 6.70892675767121809281826838397, 7.17303914609409998231984775398, 7.66401402942644815708678744089
