| L(s) = 1 | + 64·7-s − 10·9-s − 196·17-s − 64·23-s + 106·25-s + 512·31-s − 204·41-s + 640·47-s + 2.38e3·49-s − 640·63-s + 832·71-s − 276·73-s + 128·79-s − 629·81-s + 1.16e3·89-s + 476·97-s − 1.98e3·103-s − 604·113-s − 1.25e4·119-s + 2.59e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.96e3·153-s + ⋯ |
| L(s) = 1 | + 3.45·7-s − 0.370·9-s − 2.79·17-s − 0.580·23-s + 0.847·25-s + 2.96·31-s − 0.777·41-s + 1.98·47-s + 6.95·49-s − 1.27·63-s + 1.39·71-s − 0.442·73-s + 0.182·79-s − 0.862·81-s + 1.38·89-s + 0.498·97-s − 1.89·103-s − 0.502·113-s − 9.66·119-s + 1.95·121-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 + 1.03·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.088327381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.088327381\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 106 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2598 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 98 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 5974 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 19194 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 256 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 92842 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 102 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 71398 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 320 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 291978 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 244294 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 49466 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 296822 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 416 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 138 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 64 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 989910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 582 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 238 T + p^{3} T^{2} )^{2} \) |
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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
−13.67822346935608087163089189263, −12.42089260740481076117081928986, −11.83041443380580924483506319650, −11.57583110313415464142731708735, −11.12822947956863285704652408164, −10.70855406300588520073910657282, −10.33437900698030338384198933431, −9.191583735928561591017020860505, −8.679873894342876700325528142585, −8.255874575714206297938088971032, −8.089192463011981164000115636422, −7.20768799973568294427327842474, −6.62205745988845873508948989686, −5.75525133058376162544666269319, −4.92132557573469814446618530506, −4.58601960856081079575403047043, −4.23644807004496027662100866424, −2.48725669430463390720141254769, −2.01909209413364677560323945833, −0.987822623787148991049122495485,
0.987822623787148991049122495485, 2.01909209413364677560323945833, 2.48725669430463390720141254769, 4.23644807004496027662100866424, 4.58601960856081079575403047043, 4.92132557573469814446618530506, 5.75525133058376162544666269319, 6.62205745988845873508948989686, 7.20768799973568294427327842474, 8.089192463011981164000115636422, 8.255874575714206297938088971032, 8.679873894342876700325528142585, 9.191583735928561591017020860505, 10.33437900698030338384198933431, 10.70855406300588520073910657282, 11.12822947956863285704652408164, 11.57583110313415464142731708735, 11.83041443380580924483506319650, 12.42089260740481076117081928986, 13.67822346935608087163089189263
