| L(s) = 1 | + 32·7-s − 18·9-s − 72·17-s + 512·23-s + 52·25-s + 160·31-s + 872·41-s + 448·47-s − 316·49-s − 576·63-s + 4.09e3·71-s − 1.32e3·73-s − 992·79-s + 243·81-s − 1.06e3·89-s − 2.44e3·97-s + 3.42e3·103-s + 456·113-s − 2.30e3·119-s − 1.36e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.29e3·153-s + ⋯ |
| L(s) = 1 | + 1.72·7-s − 2/3·9-s − 1.02·17-s + 4.64·23-s + 0.415·25-s + 0.926·31-s + 3.32·41-s + 1.39·47-s − 0.921·49-s − 1.15·63-s + 6.84·71-s − 2.11·73-s − 1.41·79-s + 1/3·81-s − 1.26·89-s − 2.55·97-s + 3.27·103-s + 0.379·113-s − 1.77·119-s − 1.02·121-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 + 0.684·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.978498188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.978498188\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 18614 T^{4} - 52 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 16 T + 542 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 124 p T^{2} + 3795254 T^{4} + 124 p^{7} T^{6} + p^{12} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 4532 T^{2} + 10475286 T^{4} - 4532 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 36 T + 6822 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 11532 T^{2} + 65783830 T^{4} - 11532 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 256 T + 39886 T^{2} - 256 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 52308 T^{2} + 1484422166 T^{4} - 52308 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 80 T + 26030 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 186708 T^{2} + 13784867446 T^{4} - 186708 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 436 T + 102166 T^{2} - 436 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 57900 T^{2} + 11793718966 T^{4} - 57900 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 224 T + 179422 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 442740 T^{2} + 87795274358 T^{4} - 442740 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 305782 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 348986 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 140620 T^{2} + 86210675286 T^{4} - 140620 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2048 T + 1763566 T^{2} - 2048 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 660 T + 407702 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 496 T + 323534 T^{2} + 496 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1659916 T^{2} + 1244512983830 T^{4} - 1659916 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 532 T + 1467382 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1220 T + 586694 T^{2} + 1220 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.948919490273926557684207758723, −7.41307558194206554426927431279, −7.27150192568504865621404658773, −7.05777120539566782326530413174, −6.85387930554797081692058598828, −6.56490473534773537384508999478, −6.26597809018203429468873452697, −6.03312912555073074319391660476, −5.49833039472147976549957587842, −5.47240840413814229215303951401, −4.96531360452056762308303483580, −4.94694672132284673435557998511, −4.82182598867083594077417838878, −4.30823829320853091540522653307, −4.22144921170111837608413568024, −3.71392575480308516797721506940, −3.36036134816789442073500953724, −2.88471657120935676578383145642, −2.72957511084558171217329710992, −2.42482035271289104433297996357, −2.15848546524374831459178904879, −1.32174349401729479741596330884, −1.27693681370273272238060629301, −0.822364584531941637277034558000, −0.49110437182719285386602376106,
0.49110437182719285386602376106, 0.822364584531941637277034558000, 1.27693681370273272238060629301, 1.32174349401729479741596330884, 2.15848546524374831459178904879, 2.42482035271289104433297996357, 2.72957511084558171217329710992, 2.88471657120935676578383145642, 3.36036134816789442073500953724, 3.71392575480308516797721506940, 4.22144921170111837608413568024, 4.30823829320853091540522653307, 4.82182598867083594077417838878, 4.94694672132284673435557998511, 4.96531360452056762308303483580, 5.47240840413814229215303951401, 5.49833039472147976549957587842, 6.03312912555073074319391660476, 6.26597809018203429468873452697, 6.56490473534773537384508999478, 6.85387930554797081692058598828, 7.05777120539566782326530413174, 7.27150192568504865621404658773, 7.41307558194206554426927431279, 7.948919490273926557684207758723
