| L(s) = 1 | − 42·5-s + 686·7-s − 7.42e3·11-s + 1.18e4·13-s − 1.57e4·17-s + 2.66e4·19-s − 3.26e4·23-s − 1.23e5·25-s + 1.58e5·29-s − 1.80e5·31-s − 2.88e4·35-s − 4.58e4·37-s + 3.21e5·41-s + 1.02e6·43-s − 1.66e6·47-s + 3.52e5·49-s + 4.10e5·53-s + 3.11e5·55-s − 1.70e6·59-s − 5.47e5·61-s − 4.96e5·65-s − 2.59e6·67-s − 4.12e6·71-s − 8.00e6·73-s − 5.09e6·77-s + 2.47e6·79-s + 9.90e6·83-s + ⋯ |
| L(s) = 1 | − 0.150·5-s + 0.755·7-s − 1.68·11-s + 1.49·13-s − 0.779·17-s + 0.890·19-s − 0.559·23-s − 1.57·25-s + 1.20·29-s − 1.08·31-s − 0.113·35-s − 0.148·37-s + 0.729·41-s + 1.96·43-s − 2.34·47-s + 3/7·49-s + 0.378·53-s + 0.252·55-s − 1.07·59-s − 0.308·61-s − 0.224·65-s − 1.05·67-s − 1.36·71-s − 2.40·73-s − 1.27·77-s + 0.563·79-s + 1.90·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 42 T + 24986 p T^{2} + 42 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7428 T + 46542982 T^{2} + 7428 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 70 p^{2} T + 160198410 T^{2} - 70 p^{9} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 15792 T + 526157566 T^{2} + 15792 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 26614 T + 1962247086 T^{2} - 26614 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 32640 T - 991808882 T^{2} + 32640 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 158016 T + 39988772806 T^{2} - 158016 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 180740 T + 38484320958 T^{2} + 180740 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 45824 T - 18085535274 T^{2} + 45824 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 321720 T + 181439440606 T^{2} - 321720 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1023868 T + 671194246566 T^{2} - 1023868 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1665972 T + 1675477834078 T^{2} + 1665972 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 410628 T + 2334205080574 T^{2} - 410628 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1702134 T + 4026188129518 T^{2} + 1702134 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 547526 T + 5609323300002 T^{2} + 547526 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2590616 T + 4654840470246 T^{2} + 2590616 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4129272 T + 22218672158062 T^{2} + 4129272 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8008868 T + 520866105078 p T^{2} + 8008868 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2470456 T - 13654752819234 T^{2} - 2470456 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9900786 T + 68835957963214 T^{2} - 9900786 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15423492 T + 143317325773078 T^{2} + 15423492 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17377472 T + 164552259333822 T^{2} + 17377472 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.56465237177144979119831109850, −10.30046996420870374637162347934, −9.497817970646493273006464450478, −9.171669599220728907859902151180, −8.379402057985717868188270064799, −8.208345238575597404676964245942, −7.56705615372523614415459810711, −7.41502371030172150145270462484, −6.37919982814729788929000249761, −6.03307864589437708312328216421, −5.39035888542135971376642949096, −5.04540389190013401059660538677, −4.06205422636702563107491368285, −4.02894797221747031878569954656, −2.84191344202672135354304369289, −2.63905649865190369799623957913, −1.53070495589636711504091389940, −1.34345509340296335783139924662, 0, 0,
1.34345509340296335783139924662, 1.53070495589636711504091389940, 2.63905649865190369799623957913, 2.84191344202672135354304369289, 4.02894797221747031878569954656, 4.06205422636702563107491368285, 5.04540389190013401059660538677, 5.39035888542135971376642949096, 6.03307864589437708312328216421, 6.37919982814729788929000249761, 7.41502371030172150145270462484, 7.56705615372523614415459810711, 8.208345238575597404676964245942, 8.379402057985717868188270064799, 9.171669599220728907859902151180, 9.497817970646493273006464450478, 10.30046996420870374637162347934, 10.56465237177144979119831109850
