| L(s) = 1 | − 50·5-s + 158·9-s + 1.87e3·25-s − 2.39e3·29-s − 964·41-s − 7.90e3·45-s − 1.92e3·49-s + 8.15e3·61-s + 1.84e4·81-s + 8.64e3·89-s − 1.10e4·101-s + 4.46e4·109-s + 2.92e4·121-s − 6.25e4·125-s + 127-s + 131-s + 137-s + 139-s + 1.19e5·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5.71e4·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2·5-s + 1.95·9-s + 3·25-s − 2.84·29-s − 0.573·41-s − 3.90·45-s − 0.800·49-s + 2.19·61-s + 2.80·81-s + 1.09·89-s − 1.08·101-s + 3.75·109-s + 2·121-s − 4·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s + 5.69·145-s + 4.50e−5·149-s + 4.38e−5·151-s + 4.05e−5·157-s + 3.76e−5·163-s + 3.58e−5·167-s + 2·169-s + 3.34e−5·173-s + 3.12e−5·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.327950467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.327950467\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 158 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 1922 T^{2} + p^{8} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 211202 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1198 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 482 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 2519518 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9618242 T^{2} + p^{8} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4078 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 20249758 T^{2} + p^{8} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 30884638 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4322 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
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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
−13.90319015114450438974878431278, −13.03764634982291988727510504523, −12.80998375142703587648468084982, −12.45484572807308830484850236754, −11.52760739255604168386460459224, −11.42982341193472034988538482564, −10.70725655009193865283633367657, −10.08193690129939261647473974363, −9.472106764632052052471782868410, −8.757986258998767481688099955150, −8.052221660603498424879002092169, −7.38946375002547291939714172848, −7.27155414958287336273214312607, −6.51004309971067595480141956007, −5.26284193809888240197257228261, −4.52776064918683640183755519102, −3.88854060184822673985402133660, −3.44936722688872881767358353743, −1.82812879793037961166209991285, −0.60369903617915881898722533219,
0.60369903617915881898722533219, 1.82812879793037961166209991285, 3.44936722688872881767358353743, 3.88854060184822673985402133660, 4.52776064918683640183755519102, 5.26284193809888240197257228261, 6.51004309971067595480141956007, 7.27155414958287336273214312607, 7.38946375002547291939714172848, 8.052221660603498424879002092169, 8.757986258998767481688099955150, 9.472106764632052052471782868410, 10.08193690129939261647473974363, 10.70725655009193865283633367657, 11.42982341193472034988538482564, 11.52760739255604168386460459224, 12.45484572807308830484850236754, 12.80998375142703587648468084982, 13.03764634982291988727510504523, 13.90319015114450438974878431278
