| L(s) = 1 | − i·2-s − 4-s + 2i·7-s + i·8-s + 4·11-s − i·13-s + 2·14-s + 16-s + 6·19-s − 4i·22-s − 4i·23-s − 26-s − 2i·28-s − 8·29-s − 2·31-s − i·32-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.755i·7-s + 0.353i·8-s + 1.20·11-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s + 1.37·19-s − 0.852i·22-s − 0.834i·23-s − 0.196·26-s − 0.377i·28-s − 1.48·29-s − 0.359·31-s − 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.981778787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.981778787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.201715606930708252025037642233, −7.23854486688122120611865192190, −6.59951484815488178626689415255, −5.50562291979258512182371338605, −5.28782373922338305537407371163, −4.00212743948860978137508466672, −3.58487593539234385744396251027, −2.54774098240635433239818409795, −1.77433640158321144240438325485, −0.68105099931596257592621609714,
0.884521569734502184556520834614, 1.77393214030762998308744781617, 3.35807442193009513728454564044, 3.77886063266321019197301658509, 4.67415203493844251435354007373, 5.42678464495939484324141187873, 6.17813753322531809211104249985, 6.95309731478338360853701405256, 7.37122253526833356456622698365, 8.046351333288953364648310047215
