| L(s) = 1 | + 4·2-s − 8·3-s + 12·4-s − 32·6-s + 32·8-s − 6·9-s − 96·12-s − 84·13-s + 80·16-s − 24·18-s − 256·24-s − 146·25-s − 336·26-s + 392·27-s − 44·29-s + 592·31-s + 192·32-s − 72·36-s + 672·39-s − 636·41-s − 368·47-s − 640·48-s − 270·49-s − 584·50-s − 1.00e3·52-s + 1.56e3·54-s − 176·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.53·3-s + 3/2·4-s − 2.17·6-s + 1.41·8-s − 2/9·9-s − 2.30·12-s − 1.79·13-s + 5/4·16-s − 0.314·18-s − 2.17·24-s − 1.16·25-s − 2.53·26-s + 2.79·27-s − 0.281·29-s + 3.42·31-s + 1.06·32-s − 1/3·36-s + 2.75·39-s − 2.42·41-s − 1.14·47-s − 1.92·48-s − 0.787·49-s − 1.65·50-s − 2.68·52-s + 3.95·54-s − 0.398·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6470255950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6470255950\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + 4 T + p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 146 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 270 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2434 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 42 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9410 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5294 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 296 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 25490 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 318 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 59070 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 184 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 289330 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 500 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 453026 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 188750 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 210 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 182466 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 184402 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 744338 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 277410 T^{2} + p^{6} T^{4} \) |
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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
−9.920652619416608650123012330460, −9.645582813740595956351871985673, −8.785618222073313657485952162395, −8.414914404301837080069977099496, −7.78812429942853980530227384096, −7.66289569197074928551556513183, −6.76183133325775121818288100102, −6.65337618441228491502437275743, −6.09160097720698582865031022373, −6.02990543612263850384814116027, −5.17927784856347654626433937946, −5.11762209569810079278331113744, −4.59432108303203310397189840610, −4.47183464344368122355556214479, −3.30210050984986435809655227221, −3.17232301839513480747010235818, −2.50299556306513477633871717902, −1.96198321738219333673548290917, −1.06248712243162700266691420609, −0.18225839287185132476268210778,
0.18225839287185132476268210778, 1.06248712243162700266691420609, 1.96198321738219333673548290917, 2.50299556306513477633871717902, 3.17232301839513480747010235818, 3.30210050984986435809655227221, 4.47183464344368122355556214479, 4.59432108303203310397189840610, 5.11762209569810079278331113744, 5.17927784856347654626433937946, 6.02990543612263850384814116027, 6.09160097720698582865031022373, 6.65337618441228491502437275743, 6.76183133325775121818288100102, 7.66289569197074928551556513183, 7.78812429942853980530227384096, 8.414914404301837080069977099496, 8.785618222073313657485952162395, 9.645582813740595956351871985673, 9.920652619416608650123012330460
