| L(s) = 1 | + 2.56i·2-s − 4.56·4-s + 0.438i·7-s − 6.56i·8-s + 1.56·11-s − i·13-s − 1.12·14-s + 7.68·16-s + 1.56i·17-s + 5.12·19-s + 4i·22-s − 2.43i·23-s + 2.56·26-s − 2i·28-s − 7.12·29-s + ⋯ |
| L(s) = 1 | + 1.81i·2-s − 2.28·4-s + 0.165i·7-s − 2.31i·8-s + 0.470·11-s − 0.277i·13-s − 0.300·14-s + 1.92·16-s + 0.378i·17-s + 1.17·19-s + 0.852i·22-s − 0.508i·23-s + 0.502·26-s − 0.377i·28-s − 1.32·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.290072691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.290072691\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 2.56iT - 2T^{2} \) |
| 7 | \( 1 - 0.438iT - 7T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 - 1.56iT - 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 2.43iT - 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 + 3.56T + 41T^{2} \) |
| 43 | \( 1 - 3.12iT - 43T^{2} \) |
| 47 | \( 1 - 11.1iT - 47T^{2} \) |
| 53 | \( 1 - 4.68iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 - 11.3iT - 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 - 16.4iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 16.9iT - 97T^{2} \) |
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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
−8.898942044970303426818850225193, −8.253999989349499851125286395263, −7.60508644659127995401246774764, −6.92840829989018833498330826827, −6.17988375769006573427520085754, −5.60892899511881910138640671440, −4.78567175008528779806765043037, −4.04682261442079229288179408400, −2.94216275208139638414627137435, −1.15221598377197832761047421865,
0.48471499608281296058666312327, 1.57610397372551341443888165442, 2.43892221276140374346724149099, 3.49810365402455767126463321208, 3.94231530562204292640503423490, 4.97694620584823684530548837001, 5.65573338457280430693164559778, 6.95052373242292595834421700705, 7.72614416893716855978414350412, 8.818069352856435234274598787472
