L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 2·11-s − 2·13-s − 15-s + 6·19-s + 21-s − 23-s + 25-s − 27-s − 8·29-s + 10·31-s + 2·33-s − 35-s − 10·37-s + 2·39-s − 10·41-s − 2·43-s + 45-s + 2·47-s + 49-s − 6·53-s − 2·55-s − 6·57-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.554·13-s − 0.258·15-s + 1.37·19-s + 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 1.79·31-s + 0.348·33-s − 0.169·35-s − 1.64·37-s + 0.320·39-s − 1.56·41-s − 0.304·43-s + 0.149·45-s + 0.291·47-s + 1/7·49-s − 0.824·53-s − 0.269·55-s − 0.794·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.32812549995219, −14.47472327280316, −13.95292877095733, −13.52217327583140, −13.00716079656807, −12.48401426925919, −11.85028655546934, −11.59158676958974, −10.85051847565471, −10.12867386860965, −10.01863658557772, −9.403552073573905, −8.756856468630896, −8.013784388018286, −7.544479611130254, −6.799994582138089, −6.516349196916861, −5.647094695278564, −5.163514207864542, −4.914294199498455, −3.806879422311120, −3.322791707415992, −2.490857826201901, −1.822951266727992, −0.8989424922409106, 0,
0.8989424922409106, 1.822951266727992, 2.490857826201901, 3.322791707415992, 3.806879422311120, 4.914294199498455, 5.163514207864542, 5.647094695278564, 6.516349196916861, 6.799994582138089, 7.544479611130254, 8.013784388018286, 8.756856468630896, 9.403552073573905, 10.01863658557772, 10.12867386860965, 10.85051847565471, 11.59158676958974, 11.85028655546934, 12.48401426925919, 13.00716079656807, 13.52217327583140, 13.95292877095733, 14.47472327280316, 15.32812549995219