| L(s) = 1 | + (−0.366 + 1.36i)2-s + i·3-s + (−1.73 − i)4-s − i·5-s + (−1.36 − 0.366i)6-s + 1.26·7-s + (2 − 1.99i)8-s − 9-s + (1.36 + 0.366i)10-s − 1.26i·11-s + (1 − 1.73i)12-s − i·13-s + (−0.464 + 1.73i)14-s + 15-s + (1.99 + 3.46i)16-s − 4·17-s + ⋯ |
| L(s) = 1 | + (−0.258 + 0.965i)2-s + 0.577i·3-s + (−0.866 − 0.5i)4-s − 0.447i·5-s + (−0.557 − 0.149i)6-s + 0.479·7-s + (0.707 − 0.707i)8-s − 0.333·9-s + (0.431 + 0.115i)10-s − 0.382i·11-s + (0.288 − 0.499i)12-s − 0.277i·13-s + (−0.124 + 0.462i)14-s + 0.258·15-s + (0.499 + 0.866i)16-s − 0.970·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7064844796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7064844796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.366 - 1.36i)T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 1.26iT - 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 2.73iT - 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 8.92iT - 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 7.46iT - 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 5.46iT - 43T^{2} \) |
| 47 | \( 1 + 5.26T + 47T^{2} \) |
| 53 | \( 1 + 8.39iT - 53T^{2} \) |
| 59 | \( 1 + 9.66iT - 59T^{2} \) |
| 61 | \( 1 - 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 1.80iT - 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 8.73iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 7.46T + 97T^{2} \) |
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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
−9.289059376950692355812803433991, −8.405861184473153393896363682852, −7.954296134868222799170746826946, −6.97066546018841261984810959228, −5.88629558276592332916677587575, −5.41391005453253864284616807636, −4.38831207021029009336823300371, −3.77060443452756139393131604342, −1.98705410388537603673679862340, −0.30578429124910456884843301090,
1.42999794106615314411448265830, 2.29861345109544461781731957654, 3.28479381709122669931245149878, 4.39960246930139254560480992651, 5.19790737021248705523940474248, 6.46385023775830908962635605748, 7.29346529968924520380274813452, 7.999426066761603616713139848055, 8.932184486623529107288205699002, 9.440992468666269743124298656926
