L(s) = 1 | − i·3-s − i·7-s − 9-s + 4i·13-s − 3i·17-s + 4·19-s − 21-s + i·23-s + i·27-s + 3·29-s − 7·31-s + 11i·37-s + 4·39-s − 9·41-s + 4i·43-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 1.10i·13-s − 0.727i·17-s + 0.917·19-s − 0.218·21-s + 0.208i·23-s + 0.192i·27-s + 0.557·29-s − 1.25·31-s + 1.80i·37-s + 0.640·39-s − 1.40·41-s + 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.263001624\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.263001624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 - 11iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.040137664935669033519495330327, −7.20947023688811931026935980631, −6.86744077953363933208801249477, −6.12519237821493111299049909418, −5.19466808682169132593165654582, −4.61779025839317162789063680333, −3.61413455421682920703527380155, −2.89063515307166714863420891091, −1.84229464301911745175299685249, −1.05126687710298547130224223050,
0.33292646355384807422317182810, 1.68030843271921561787969637683, 2.72775880597307823600690001192, 3.46039571636784232692857879691, 4.14218874847360148194961915881, 5.18847283552332095693484090054, 5.55711147808697817417414054391, 6.26725986814166630022437696395, 7.31086571659068924773226946475, 7.78292729899159681713615993103