| L(s) = 1 | + 3-s + 2·5-s + 7-s + 3·9-s − 4·13-s + 2·15-s + 3·17-s + 7·19-s + 21-s + 6·23-s + 3·25-s + 8·27-s + 3·29-s − 31-s + 2·35-s + 13·37-s − 4·39-s − 8·43-s + 6·45-s + 6·47-s − 5·49-s + 3·51-s + 9·53-s + 7·57-s − 6·59-s + 5·61-s + 3·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 9-s − 1.10·13-s + 0.516·15-s + 0.727·17-s + 1.60·19-s + 0.218·21-s + 1.25·23-s + 3/5·25-s + 1.53·27-s + 0.557·29-s − 0.179·31-s + 0.338·35-s + 2.13·37-s − 0.640·39-s − 1.21·43-s + 0.894·45-s + 0.875·47-s − 5/7·49-s + 0.420·51-s + 1.23·53-s + 0.927·57-s − 0.781·59-s + 0.640·61-s + 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.731724823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.731724823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 13 T + 108 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 120 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 174 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.70837678115078068468307058856, −7.42443082449119976594260592075, −7.30687375085709500167844297368, −6.92959878911087175955034026628, −6.51873327884104117883541925431, −6.27805625433416030118669190540, −5.52347934693889609092478165812, −5.50129488730413744253595469089, −5.23966751831607788958725181011, −4.58740077518248339102837076369, −4.46125188742221254239961649807, −4.23678192317622752818382017896, −3.28622364632750407140107091290, −3.22648502535833563942470720301, −2.84304178277390745874845676440, −2.44858886070024189818870757787, −1.99662104973823714369183526767, −1.33828576466811018476823849755, −1.18762964365922454785456621733, −0.63689160361149013397206052072,
0.63689160361149013397206052072, 1.18762964365922454785456621733, 1.33828576466811018476823849755, 1.99662104973823714369183526767, 2.44858886070024189818870757787, 2.84304178277390745874845676440, 3.22648502535833563942470720301, 3.28622364632750407140107091290, 4.23678192317622752818382017896, 4.46125188742221254239961649807, 4.58740077518248339102837076369, 5.23966751831607788958725181011, 5.50129488730413744253595469089, 5.52347934693889609092478165812, 6.27805625433416030118669190540, 6.51873327884104117883541925431, 6.92959878911087175955034026628, 7.30687375085709500167844297368, 7.42443082449119976594260592075, 7.70837678115078068468307058856
