| L(s) = 1 | + (1.36 − 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (2.36 + 2.36i)5-s + (−0.633 + 2.36i)6-s + (6.73 + 1.80i)7-s + (1.99 − 2i)8-s + (−1.5 − 2.59i)9-s + (4.09 + 2.36i)10-s + (−0.339 − 1.26i)11-s + 3.46i·12-s + (−11.2 − 6.5i)13-s + 9.85·14-s + (−5.59 + 1.5i)15-s + (1.99 − 3.46i)16-s + (−21.1 + 12.2i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.288 + 0.5i)3-s + (0.433 − 0.250i)4-s + (0.473 + 0.473i)5-s + (−0.105 + 0.394i)6-s + (0.961 + 0.257i)7-s + (0.249 − 0.250i)8-s + (−0.166 − 0.288i)9-s + (0.409 + 0.236i)10-s + (−0.0308 − 0.115i)11-s + 0.288i·12-s + (−0.866 − 0.5i)13-s + 0.704·14-s + (−0.373 + 0.100i)15-s + (0.124 − 0.216i)16-s + (−1.24 + 0.719i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.958 - 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.78867 + 0.258989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78867 + 0.258989i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.366i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 13 | \( 1 + (11.2 + 6.5i)T \) |
| good | 5 | \( 1 + (-2.36 - 2.36i)T + 25iT^{2} \) |
| 7 | \( 1 + (-6.73 - 1.80i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (0.339 + 1.26i)T + (-104. + 60.5i)T^{2} \) |
| 17 | \( 1 + (21.1 - 12.2i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-5.80 + 21.6i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (12.5 + 7.26i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (10.6 - 18.4i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-18.5 - 18.5i)T + 961iT^{2} \) |
| 37 | \( 1 + (3.96 + 14.7i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (31.9 - 8.55i)T + (1.45e3 - 840.5i)T^{2} \) |
| 43 | \( 1 + (-57.3 + 33.1i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (30 - 30i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 - 53.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-102. - 27.4i)T + (3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-41.8 - 72.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (53.5 - 14.3i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + (-8.87 + 33.1i)T + (-4.36e3 - 2.52e3i)T^{2} \) |
| 73 | \( 1 + (73.0 - 73.0i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 6.43T + 6.24e3T^{2} \) |
| 83 | \( 1 + (13.4 + 13.4i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (28.2 + 105. i)T + (-6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (47.6 - 177. i)T + (-8.14e3 - 4.70e3i)T^{2} \) |
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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
−14.39625950020616378716458192831, −13.24400145992087072539060958452, −11.98872261256751082466144089740, −10.99580675203199911181189993934, −10.17427490934116220143505482233, −8.645221100323159297636829906342, −6.93855916397456937589110878590, −5.55821373985764018857230094655, −4.44705157578227835910099073746, −2.47347525729957033560410882985,
1.97329089849788707063180145632, 4.47170930353064340930595193076, 5.56978328898653831021750454337, 7.00957231493137362623788032701, 8.132555249438515830822131769800, 9.713391699712137228467811964590, 11.28164368989945368376132337832, 12.02928754895866342049620392398, 13.23401594578632316218431739014, 13.97962434187356743128304199926
