| L(s) = 1 | + 250·5-s − 560·7-s − 3.72e3·11-s + 7.36e3·13-s − 5.94e3·17-s + 3.05e4·19-s + 2.68e4·23-s + 4.68e4·25-s + 1.56e5·29-s − 1.78e5·31-s − 1.40e5·35-s + 5.11e4·37-s + 4.34e5·41-s − 4.55e5·43-s + 1.13e5·47-s − 3.47e5·49-s + 1.98e6·53-s − 9.30e5·55-s + 3.09e6·59-s − 4.74e5·61-s + 1.84e6·65-s + 2.72e6·67-s + 6.66e6·71-s + 3.28e6·73-s + 2.08e6·77-s − 3.61e6·79-s + 8.58e6·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.617·7-s − 0.842·11-s + 0.929·13-s − 0.293·17-s + 1.02·19-s + 0.460·23-s + 3/5·25-s + 1.19·29-s − 1.07·31-s − 0.551·35-s + 0.166·37-s + 0.984·41-s − 0.874·43-s + 0.159·47-s − 0.421·49-s + 1.82·53-s − 0.753·55-s + 1.96·59-s − 0.267·61-s + 0.831·65-s + 1.10·67-s + 2.20·71-s + 0.986·73-s + 0.520·77-s − 0.823·79-s + 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.208234761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.208234761\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 80 p T + 661002 T^{2} + 80 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3720 T + 15821842 T^{2} + 3720 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7360 T + 134781498 T^{2} - 7360 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5940 T + 723049846 T^{2} + 5940 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 30544 T + 1914529062 T^{2} - 30544 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 26880 T + 6564490894 T^{2} - 26880 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 156660 T + 19771455118 T^{2} - 156660 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 178328 T + 42111560718 T^{2} + 178328 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 51160 T + 143574346266 T^{2} - 51160 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 434640 T + 432904387762 T^{2} - 434640 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 455840 T + 592706383878 T^{2} + 455840 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 113520 T + 115488680926 T^{2} - 113520 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1981380 T + 3043052482174 T^{2} - 1981380 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 3097800 T + 7024449157138 T^{2} - 3097800 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 474332 T - 100097990802 T^{2} + 474332 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2727040 T + 3175462968582 T^{2} - 2727040 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6664080 T + 28946880026782 T^{2} - 6664080 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 3280540 T + 23401879304694 T^{2} - 3280540 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3610712 T + 41571324699054 T^{2} + 3610712 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8586240 T + 57412626689254 T^{2} - 8586240 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10807080 T + 77005715601058 T^{2} - 10807080 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3695660 T + 164718762906342 T^{2} + 3695660 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.39385961571922018837008223595, −11.20725244809145585710773285010, −10.39334566743760666707236673258, −10.26485431557258817507846909063, −9.502081305757467334706417310797, −9.327465825106601926618408030791, −8.394917045378984764718984988406, −8.375886435660018577543998620272, −7.24261229684836159430296728851, −7.09399275057707900763681616618, −6.12145470360161612360531218951, −6.04799441510281426900683711443, −5.07455436208729248723646666085, −4.96527778090383359594709450957, −3.69590379049757568075740731680, −3.41837222259944551245449232120, −2.46336196693407088549011372172, −2.10867492554236543147837127238, −0.976766505189408700146961838512, −0.64507090979671922043338272363,
0.64507090979671922043338272363, 0.976766505189408700146961838512, 2.10867492554236543147837127238, 2.46336196693407088549011372172, 3.41837222259944551245449232120, 3.69590379049757568075740731680, 4.96527778090383359594709450957, 5.07455436208729248723646666085, 6.04799441510281426900683711443, 6.12145470360161612360531218951, 7.09399275057707900763681616618, 7.24261229684836159430296728851, 8.375886435660018577543998620272, 8.394917045378984764718984988406, 9.327465825106601926618408030791, 9.502081305757467334706417310797, 10.26485431557258817507846909063, 10.39334566743760666707236673258, 11.20725244809145585710773285010, 11.39385961571922018837008223595
