| L(s) = 1 | + 0.657i·3-s − i·5-s + 1.07·7-s + 2.56·9-s + 5.98·11-s − 3.56·13-s + 0.657·15-s + 4.56i·17-s − 4.31·19-s + 0.706i·21-s + (3.59 − 3.17i)23-s − 25-s + 3.66i·27-s + 9.42·29-s − 9.28i·31-s + ⋯ |
| L(s) = 1 | + 0.379i·3-s − 0.447i·5-s + 0.406·7-s + 0.855·9-s + 1.80·11-s − 0.989·13-s + 0.169·15-s + 1.10i·17-s − 0.990·19-s + 0.154i·21-s + (0.749 − 0.662i)23-s − 0.200·25-s + 0.704i·27-s + 1.75·29-s − 1.66i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.198i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.127353837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.127353837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-3.59 + 3.17i)T \) |
| good | 3 | \( 1 - 0.657iT - 3T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 - 5.98T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 4.56iT - 17T^{2} \) |
| 19 | \( 1 + 4.31T + 19T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 9.28iT - 31T^{2} \) |
| 37 | \( 1 - 8.93iT - 37T^{2} \) |
| 41 | \( 1 - 0.656T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 + 1.86iT - 47T^{2} \) |
| 53 | \( 1 + 1.79iT - 53T^{2} \) |
| 59 | \( 1 - 3.68iT - 59T^{2} \) |
| 61 | \( 1 + 4.08iT - 61T^{2} \) |
| 67 | \( 1 + 3.50T + 67T^{2} \) |
| 71 | \( 1 + 3.90iT - 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 7.92T + 79T^{2} \) |
| 83 | \( 1 - 3.50T + 83T^{2} \) |
| 89 | \( 1 + 5.13iT - 89T^{2} \) |
| 97 | \( 1 - 15.7iT - 97T^{2} \) |
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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
−9.306587437661837295035898789924, −8.559828525254007675345369458015, −7.84781530864641888314356540222, −6.66549448246415746337719484532, −6.33729669546083474209019233662, −4.84020455106905962187463013688, −4.46154902179431416686707453820, −3.63508993987010152287090815715, −2.12387012026574227660536033447, −1.09664010888153218031563642288,
1.05853302137433780214726886843, 2.09840006875943719599028934688, 3.27358547440666670516731022551, 4.35779400063428383571270600170, 4.99258428073424632267949931060, 6.34699557220811238705240328961, 6.92445398206178789355386304659, 7.37455263340769556338326368337, 8.477280883707624734149051531668, 9.269062945076280176615140449652
