| L(s) = 1 | − 18·3-s − 22·7-s + 243·9-s + 372·11-s − 10·13-s − 1.21e3·17-s − 1.55e3·19-s + 396·21-s − 2.65e3·23-s − 2.91e3·27-s − 1.81e3·29-s + 1.09e4·31-s − 6.69e3·33-s − 3.31e3·37-s + 180·39-s + 1.90e4·41-s − 1.22e4·43-s + 1.52e4·47-s − 8.05e3·49-s + 2.18e4·51-s − 2.07e4·53-s + 2.79e4·57-s + 1.87e4·59-s + 3.57e4·61-s − 5.34e3·63-s + 9.81e4·67-s + 4.77e4·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.169·7-s + 9-s + 0.926·11-s − 0.0164·13-s − 1.01·17-s − 0.985·19-s + 0.195·21-s − 1.04·23-s − 0.769·27-s − 0.400·29-s + 2.04·31-s − 1.07·33-s − 0.398·37-s + 0.0189·39-s + 1.77·41-s − 1.01·43-s + 1.00·47-s − 0.479·49-s + 1.17·51-s − 1.01·53-s + 1.13·57-s + 0.699·59-s + 1.22·61-s − 0.169·63-s + 2.67·67-s + 1.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.662282407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662282407\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 22 T + 8535 T^{2} + 22 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 372 T + 129898 T^{2} - 372 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 339411 T^{2} + 10 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1212 T + 2980150 T^{2} + 1212 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 1550 T + 2503623 T^{2} + 1550 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2652 T + 14404162 T^{2} + 2652 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1812 T - 9186866 T^{2} + 1812 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10918 T + 87033783 T^{2} - 10918 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3316 T + 133272078 T^{2} + 3316 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 19080 T + 322497202 T^{2} - 19080 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12262 T + 310412847 T^{2} + 12262 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 15252 T + 386209090 T^{2} - 15252 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 20784 T + 258314650 T^{2} + 20784 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18708 T + 24733646 p T^{2} - 18708 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 35734 T + 1963969491 T^{2} - 35734 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 98162 T + 5108967975 T^{2} - 98162 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 114120 T + 6722552302 T^{2} - 114120 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 109876 T + 6502978230 T^{2} + 109876 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 95536 T + 7260163422 T^{2} - 95536 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 3732 T + 856206442 T^{2} + 3732 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 93600 T + 12919274098 T^{2} - 93600 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 91886 T + 16163058963 T^{2} - 91886 p^{5} T^{3} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14126141034663471268399568920, −10.83951366071122308501968324374, −10.14010488067170052583857254451, −9.907476036414229721688906806300, −9.347051336876659156637393934239, −8.802814522501150367327816888375, −8.246606697280201622084186566960, −7.81918054324902532501659826241, −6.95186497001359970425146622003, −6.68670031865492033002782167641, −6.16848304090555940601549099858, −5.93766766863378500543244127753, −4.98975228755331158479235556278, −4.65721751478584142515012381922, −3.99314517245931336515505948092, −3.59803255438157349627712933141, −2.36548571588502417609831955605, −1.98722447477756228946855692563, −0.917724725012086060429128002088, −0.47775481965130712052192006497,
0.47775481965130712052192006497, 0.917724725012086060429128002088, 1.98722447477756228946855692563, 2.36548571588502417609831955605, 3.59803255438157349627712933141, 3.99314517245931336515505948092, 4.65721751478584142515012381922, 4.98975228755331158479235556278, 5.93766766863378500543244127753, 6.16848304090555940601549099858, 6.68670031865492033002782167641, 6.95186497001359970425146622003, 7.81918054324902532501659826241, 8.246606697280201622084186566960, 8.802814522501150367327816888375, 9.347051336876659156637393934239, 9.907476036414229721688906806300, 10.14010488067170052583857254451, 10.83951366071122308501968324374, 11.14126141034663471268399568920
