L(s) = 1 | + (0.711 + 1.86i)2-s + 4.44i·3-s + (−2.98 + 2.66i)4-s + 4.97·5-s + (−8.31 + 3.16i)6-s − 12.2i·7-s + (−7.09 − 3.68i)8-s − 10.7·9-s + (3.54 + 9.30i)10-s + 13.4i·11-s + (−11.8 − 13.2i)12-s + 14.1·13-s + (22.9 − 8.72i)14-s + 22.1i·15-s + (1.84 − 15.8i)16-s − 5.89·17-s + ⋯ |
L(s) = 1 | + (0.355 + 0.934i)2-s + 1.48i·3-s + (−0.746 + 0.665i)4-s + 0.995·5-s + (−1.38 + 0.527i)6-s − 1.75i·7-s + (−0.887 − 0.461i)8-s − 1.19·9-s + (0.354 + 0.930i)10-s + 1.22i·11-s + (−0.986 − 1.10i)12-s + 1.09·13-s + (1.63 − 0.623i)14-s + 1.47i·15-s + (0.115 − 0.993i)16-s − 0.346·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.629294 + 1.40331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.629294 + 1.40331i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.711 - 1.86i)T \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 - 4.44iT - 9T^{2} \) |
| 5 | \( 1 - 4.97T + 25T^{2} \) |
| 7 | \( 1 + 12.2iT - 49T^{2} \) |
| 11 | \( 1 - 13.4iT - 121T^{2} \) |
| 13 | \( 1 - 14.1T + 169T^{2} \) |
| 17 | \( 1 + 5.89T + 289T^{2} \) |
| 23 | \( 1 + 0.906iT - 529T^{2} \) |
| 29 | \( 1 + 10.3T + 841T^{2} \) |
| 31 | \( 1 + 43.2iT - 961T^{2} \) |
| 37 | \( 1 + 1.61T + 1.36e3T^{2} \) |
| 41 | \( 1 - 69.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 32.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 38.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 8.31T + 2.80e3T^{2} \) |
| 59 | \( 1 + 20.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 118.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 57.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 11.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 23.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 0.286iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 24.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 43.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 115.T + 9.40e3T^{2} \) |
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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
−14.74822181719587093372212541455, −13.79963319909263615918754661917, −13.06771283300753005460493259802, −10.98142586055868705943982180222, −9.963148469414577129575517954905, −9.270943798168117405637474343137, −7.56178870720052699032033913751, −6.18029446098285927461373153494, −4.68022535086379776671605372582, −3.87307261716257803183768788115,
1.58423177261008336429867314971, 2.81400489300831810842078926254, 5.72808470595039932793024412330, 6.12566859038458581874836167892, 8.450622197990576321043542634664, 9.171365861564149332752039070045, 10.92497541056863234796971560281, 11.93848127309956208454493169848, 12.78780755213878974139721285513, 13.56309194621197621046214009712