| L(s) = 1 | + 13·4-s + 50·5-s − 142·9-s − 124·11-s − 87·16-s + 722·19-s + 650·20-s + 1.87e3·25-s − 1.84e3·36-s − 1.61e3·44-s − 7.10e3·45-s + 4.80e3·49-s − 6.20e3·55-s + 1.42e4·61-s − 4.45e3·64-s + 9.38e3·76-s − 4.35e3·80-s + 1.36e4·81-s + 3.61e4·95-s + 1.76e4·99-s + 2.43e4·100-s + 4.01e4·101-s − 1.77e4·121-s + 6.25e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.812·4-s + 2·5-s − 1.75·9-s − 1.02·11-s − 0.339·16-s + 2·19-s + 13/8·20-s + 3·25-s − 1.42·36-s − 0.832·44-s − 3.50·45-s + 2·49-s − 2.04·55-s + 3.83·61-s − 1.08·64-s + 13/8·76-s − 0.679·80-s + 2.07·81-s + 4·95-s + 1.79·99-s + 2.43·100-s + 3.94·101-s − 1.21·121-s + 4·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.543337798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.543337798\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 13 T^{2} + p^{8} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 142 T^{2} + p^{8} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 62 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 52622 T^{2} + p^{8} T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 3237298 T^{2} + p^{8} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 15154382 T^{2} + p^{8} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7138 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 3364622 T^{2} + p^{8} T^{4} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 60577618 T^{2} + p^{8} T^{4} \) |
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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
−13.57334943631846293688007237545, −13.18836900876708261982688968689, −12.54643651824025999483925167147, −11.67142682040378487914843403625, −11.50513753515577700174771054630, −10.82477619658826124356223081458, −10.21498510190369114511038332773, −9.898354266982772451834703022771, −9.007168983878526367304391043211, −8.848879223948621224497156625464, −7.911359142333311285752965927687, −7.19133250618896566611253659942, −6.55050977467251966660474252264, −5.83737890371271249716412607421, −5.44855585979411610412198506732, −5.06408842290981846085975709017, −3.32073381822115920946437634512, −2.53597628261584386114806763171, −2.25572542021785236939160522503, −0.882534227291798518873035730630,
0.882534227291798518873035730630, 2.25572542021785236939160522503, 2.53597628261584386114806763171, 3.32073381822115920946437634512, 5.06408842290981846085975709017, 5.44855585979411610412198506732, 5.83737890371271249716412607421, 6.55050977467251966660474252264, 7.19133250618896566611253659942, 7.911359142333311285752965927687, 8.848879223948621224497156625464, 9.007168983878526367304391043211, 9.898354266982772451834703022771, 10.21498510190369114511038332773, 10.82477619658826124356223081458, 11.50513753515577700174771054630, 11.67142682040378487914843403625, 12.54643651824025999483925167147, 13.18836900876708261982688968689, 13.57334943631846293688007237545
