| L(s) = 1 | + 2·3-s + 2·4-s − 2·5-s − 2·7-s + 3·9-s + 4·12-s + 4·13-s − 4·15-s − 6·17-s + 4·19-s − 4·20-s − 4·21-s − 2·23-s + 3·25-s + 4·27-s − 4·28-s + 6·29-s + 10·31-s + 4·35-s + 6·36-s − 2·37-s + 8·39-s + 6·41-s + 4·43-s − 6·45-s + 12·47-s − 11·49-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4-s − 0.894·5-s − 0.755·7-s + 9-s + 1.15·12-s + 1.10·13-s − 1.03·15-s − 1.45·17-s + 0.917·19-s − 0.894·20-s − 0.872·21-s − 0.417·23-s + 3/5·25-s + 0.769·27-s − 0.755·28-s + 1.11·29-s + 1.79·31-s + 0.676·35-s + 36-s − 0.328·37-s + 1.28·39-s + 0.937·41-s + 0.609·43-s − 0.894·45-s + 1.75·47-s − 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.489142833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489142833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 124 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 73 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 97 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 234 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−11.86077193248076654356303449390, −11.22356694323448029731531008127, −10.90408889797468358163310340330, −10.38397046858691223561255767230, −9.878449378802652331564137511855, −9.388321585222944232242201741426, −8.786619330639239266159095971190, −8.485132080751759459239239063870, −8.077088563688002207896582105017, −7.43318769085195749027037335195, −7.03607347328936342367421369095, −6.51786511086534937764587802081, −6.27468451965364581486300953146, −5.37555937173621067054577784413, −4.35770431936673725435082309572, −4.13851514964343879566161932778, −3.38598750525546634169084841382, −2.73620487387102761346495515886, −2.38821824858040218890160403827, −1.12500220983655536238641794410,
1.12500220983655536238641794410, 2.38821824858040218890160403827, 2.73620487387102761346495515886, 3.38598750525546634169084841382, 4.13851514964343879566161932778, 4.35770431936673725435082309572, 5.37555937173621067054577784413, 6.27468451965364581486300953146, 6.51786511086534937764587802081, 7.03607347328936342367421369095, 7.43318769085195749027037335195, 8.077088563688002207896582105017, 8.485132080751759459239239063870, 8.786619330639239266159095971190, 9.388321585222944232242201741426, 9.878449378802652331564137511855, 10.38397046858691223561255767230, 10.90408889797468358163310340330, 11.22356694323448029731531008127, 11.86077193248076654356303449390
