| L(s) = 1 | + 16·4-s + 192·16-s − 214·19-s − 578·31-s + 397·49-s − 1.80e3·61-s + 2.04e3·64-s − 3.42e3·76-s + 2.77e3·79-s + 3.13e3·109-s − 2.66e3·121-s − 9.24e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s − 2.58·19-s − 3.34·31-s + 1.15·49-s − 3.78·61-s + 4·64-s − 5.16·76-s + 3.95·79-s + 2.75·109-s − 2·121-s − 6.69·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.413121710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.413121710\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 397 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 107 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 289 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 3023 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 153973 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 901 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 172874 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 66527 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1387 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1608263 T^{2} + p^{6} T^{4} \) |
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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
−10.56747887945356679298144533670, −10.17712589757969741038667760459, −9.411356268261820356394738401090, −8.905043386061911949033909241688, −8.730570337145880690061621206831, −7.78802604221033877643510736718, −7.70137822355891522619486783386, −7.31652492060750920544757217397, −6.67330973336487311830887867328, −6.27876200914262984881725263356, −6.15145817095084193682265900650, −5.44635669532994630972040128894, −4.99531831322089286426544307096, −4.06390949767312408526072189854, −3.73412615949949115270563548566, −3.07398771076426406441408209036, −2.43252678076250348683996330120, −1.85464672302512574374327030618, −1.69317654618269984943557928411, −0.46198086430263438703629355333,
0.46198086430263438703629355333, 1.69317654618269984943557928411, 1.85464672302512574374327030618, 2.43252678076250348683996330120, 3.07398771076426406441408209036, 3.73412615949949115270563548566, 4.06390949767312408526072189854, 4.99531831322089286426544307096, 5.44635669532994630972040128894, 6.15145817095084193682265900650, 6.27876200914262984881725263356, 6.67330973336487311830887867328, 7.31652492060750920544757217397, 7.70137822355891522619486783386, 7.78802604221033877643510736718, 8.730570337145880690061621206831, 8.905043386061911949033909241688, 9.411356268261820356394738401090, 10.17712589757969741038667760459, 10.56747887945356679298144533670
