| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 4·11-s + 13-s − 15-s + 6·17-s + 8·19-s + 21-s − 4·23-s + 25-s − 27-s + 2·29-s − 4·31-s − 4·33-s − 35-s − 10·37-s − 39-s + 6·41-s + 8·43-s + 45-s − 8·47-s + 49-s − 6·51-s + 2·53-s + 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 1.45·17-s + 1.83·19-s + 0.218·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.169·35-s − 1.64·37-s − 0.160·39-s + 0.937·41-s + 1.21·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.274·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.659538651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.659538651\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.82650380426827, −13.63525387715903, −12.76022455244472, −12.18738366738409, −12.10975943691882, −11.54685944489840, −10.88932195460514, −10.44101227735823, −9.818663843356319, −9.464759960509466, −9.151823468618894, −8.330404201066190, −7.741278672475212, −7.192601172665702, −6.769162215656968, −6.052654126416631, −5.700131460549685, −5.304179211296716, −4.530171200524342, −3.835501568216892, −3.372923288989461, −2.804298172635379, −1.674480339591219, −1.333447476689690, −0.5850185630668809,
0.5850185630668809, 1.333447476689690, 1.674480339591219, 2.804298172635379, 3.372923288989461, 3.835501568216892, 4.530171200524342, 5.304179211296716, 5.700131460549685, 6.052654126416631, 6.769162215656968, 7.192601172665702, 7.741278672475212, 8.330404201066190, 9.151823468618894, 9.464759960509466, 9.818663843356319, 10.44101227735823, 10.88932195460514, 11.54685944489840, 12.10975943691882, 12.18738366738409, 12.76022455244472, 13.63525387715903, 13.82650380426827
