| L(s) = 1 | + 2·3-s − 4·5-s + 3·9-s − 4·11-s − 4·13-s − 8·15-s + 4·17-s − 8·23-s + 4·25-s + 4·27-s − 12·29-s − 8·33-s − 12·37-s − 8·39-s − 4·41-s + 8·43-s − 12·45-s − 8·47-s − 12·49-s + 8·51-s − 12·53-s + 16·55-s − 8·59-s − 12·61-s + 16·65-s − 16·69-s + 8·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s + 0.970·17-s − 1.66·23-s + 4/5·25-s + 0.769·27-s − 2.22·29-s − 1.39·33-s − 1.97·37-s − 1.28·39-s − 0.624·41-s + 1.21·43-s − 1.78·45-s − 1.16·47-s − 1.71·49-s + 1.12·51-s − 1.64·53-s + 2.15·55-s − 1.04·59-s − 1.53·61-s + 1.98·65-s − 1.92·69-s + 0.923·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2359296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 126 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 32 T + 412 T^{2} - 32 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.262135964023016316833285959871, −8.823163930775301695731638374761, −8.067487474113410586469825652917, −8.011713369281578914986190302397, −7.73240308159423135274188229231, −7.67751568722356275156199791003, −7.04474590870111194820967875618, −6.65630761008210765367080581358, −5.90787219905264460499837571541, −5.48776283581282269060039661630, −4.88273444971697314366506320041, −4.63911921690131382495963156357, −3.94765746651364394631287618129, −3.64256683552517482307841969869, −3.24006328235972876376499831973, −2.85874701087795609631882066196, −1.93204876129471783406073253021, −1.74841003013048343609082108979, 0, 0,
1.74841003013048343609082108979, 1.93204876129471783406073253021, 2.85874701087795609631882066196, 3.24006328235972876376499831973, 3.64256683552517482307841969869, 3.94765746651364394631287618129, 4.63911921690131382495963156357, 4.88273444971697314366506320041, 5.48776283581282269060039661630, 5.90787219905264460499837571541, 6.65630761008210765367080581358, 7.04474590870111194820967875618, 7.67751568722356275156199791003, 7.73240308159423135274188229231, 8.011713369281578914986190302397, 8.067487474113410586469825652917, 8.823163930775301695731638374761, 9.262135964023016316833285959871
