L(s) = 1 | + 12·3-s + 90·9-s − 40·11-s − 120·17-s − 320·25-s + 540·27-s − 480·33-s − 552·41-s − 432·43-s − 832·49-s − 1.44e3·51-s − 1.36e3·59-s − 864·67-s − 240·73-s − 3.84e3·75-s + 2.83e3·81-s − 952·83-s − 1.56e3·89-s − 2.28e3·97-s − 3.60e3·99-s − 944·107-s − 4.20e3·113-s − 4.18e3·121-s − 6.62e3·123-s + 127-s − 5.18e3·129-s + 131-s + ⋯ |
L(s) = 1 | + 2.30·3-s + 10/3·9-s − 1.09·11-s − 1.71·17-s − 2.55·25-s + 3.84·27-s − 2.53·33-s − 2.10·41-s − 1.53·43-s − 2.42·49-s − 3.95·51-s − 3.00·59-s − 1.57·67-s − 0.384·73-s − 5.91·75-s + 35/9·81-s − 1.25·83-s − 1.85·89-s − 2.38·97-s − 3.65·99-s − 0.852·107-s − 3.49·113-s − 3.14·121-s − 4.85·123-s + 0.000698·127-s − 3.53·129-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{4} \) |
good | 5 | $D_4\times C_2$ | \( 1 + 64 p T^{2} + 1986 p^{2} T^{4} + 64 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 832 T^{2} + 7746 p^{2} T^{4} + 832 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 20 T + 2690 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8428 T^{2} + 27382614 T^{4} + 8428 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 60 T + 10334 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 12750 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 7628 T^{2} - 37861626 T^{4} + 7628 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 21056 T^{2} + 1079985426 T^{4} + 21056 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 75424 T^{2} + 3008245506 T^{4} + 75424 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 106492 T^{2} + 7210762134 T^{4} + 106492 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 276 T + 155086 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 216 T + 146478 T^{2} + 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 132332 T^{2} + 9563109414 T^{4} + 132332 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 560768 T^{2} + 122848689714 T^{4} + 560768 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 680 T + 526070 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 345244 T^{2} + 119910783606 T^{4} + 345244 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 432 T + 416982 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 1199084 T^{2} + 615208348806 T^{4} + 1199084 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 120 T + 707906 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 1369696 T^{2} + 864453694146 T^{4} + 1369696 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 476 T + 1155218 T^{2} + 476 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 780 T + 1357238 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1140 T + 2147654 T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.88076864187030466256985983582, −6.51609326640480561745353792290, −6.44128998051250685127705341066, −6.21905279632793005240688559935, −6.19430345558652033127649799442, −5.49511109386720359164372024274, −5.37787091495853732332074130301, −5.32371925869689798462246958153, −5.11708349830427572788607573639, −4.49059041340007990954171567411, −4.40697523297324033496142007851, −4.38185286182576214168712445325, −4.31426973654103408089355108814, −3.67743230845008799913238868691, −3.52716611123759888175987561333, −3.32173941682719244357383633908, −3.30815850508713374058001871441, −2.70541137801708262113865156073, −2.62964016922666933137456693574, −2.43061232492013199318325766068, −2.28384009029328334797506691889, −1.63259823996904915562747471578, −1.61220443836920412881849349673, −1.40011985458388851642270099713, −1.33046816970863279326688000088, 0, 0, 0, 0,
1.33046816970863279326688000088, 1.40011985458388851642270099713, 1.61220443836920412881849349673, 1.63259823996904915562747471578, 2.28384009029328334797506691889, 2.43061232492013199318325766068, 2.62964016922666933137456693574, 2.70541137801708262113865156073, 3.30815850508713374058001871441, 3.32173941682719244357383633908, 3.52716611123759888175987561333, 3.67743230845008799913238868691, 4.31426973654103408089355108814, 4.38185286182576214168712445325, 4.40697523297324033496142007851, 4.49059041340007990954171567411, 5.11708349830427572788607573639, 5.32371925869689798462246958153, 5.37787091495853732332074130301, 5.49511109386720359164372024274, 6.19430345558652033127649799442, 6.21905279632793005240688559935, 6.44128998051250685127705341066, 6.51609326640480561745353792290, 6.88076864187030466256985983582