| L(s) = 1 | − 54·3-s + 906·7-s + 2.18e3·9-s + 6.34e3·11-s + 6.69e3·13-s + 3.13e4·17-s + 5.64e3·19-s − 4.89e4·21-s − 3.49e4·23-s − 7.87e4·27-s − 7.97e4·29-s + 1.52e5·31-s − 3.42e5·33-s + 1.68e5·37-s − 3.61e5·39-s − 7.61e5·41-s − 3.86e5·43-s + 7.24e5·47-s − 1.00e6·49-s − 1.69e6·51-s + 1.86e6·53-s − 3.04e5·57-s − 2.11e6·59-s + 7.87e5·61-s + 1.98e6·63-s + 1.84e5·67-s + 1.88e6·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.998·7-s + 9-s + 1.43·11-s + 0.844·13-s + 1.54·17-s + 0.188·19-s − 1.15·21-s − 0.599·23-s − 0.769·27-s − 0.607·29-s + 0.920·31-s − 1.65·33-s + 0.546·37-s − 0.975·39-s − 1.72·41-s − 0.741·43-s + 1.01·47-s − 1.21·49-s − 1.78·51-s + 1.72·53-s − 0.217·57-s − 1.33·59-s + 0.444·61-s + 0.998·63-s + 0.0748·67-s + 0.692·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.248894312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.248894312\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 906 T + 1824070 T^{2} - 906 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 6342 T + 40872558 T^{2} - 6342 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6690 T + 130335434 T^{2} - 6690 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 31306 T + 852058730 T^{2} - 31306 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5640 T + 1303903478 T^{2} - 5640 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 34976 T + 5940869438 T^{2} + 34976 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 79758 T + 10563369034 T^{2} + 79758 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 152708 T + 2814513438 T^{2} - 152708 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 168318 T + 196861110922 T^{2} - 168318 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 761400 T + 521619097662 T^{2} + 761400 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 386616 T + 538164622678 T^{2} + 386616 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 724604 T + 1083517577630 T^{2} - 724604 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1864398 T + 3106081049650 T^{2} - 1864398 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2111178 T + 3567272493534 T^{2} + 2111178 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 787444 T + 6351476067726 T^{2} - 787444 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 184164 T - 1085128611530 T^{2} - 184164 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1036860 T + 7386973705582 T^{2} - 1036860 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 217428 T + 4721903347090 T^{2} + 217428 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4052436 T + 41254664174942 T^{2} + 4052436 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 5571064 T + 47327900804678 T^{2} - 5571064 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 11647740 T + 115644497697558 T^{2} - 11647740 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 28937952 T + 362085230604802 T^{2} - 28937952 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.63838633763956887465697936871, −10.54608039260938214461443661570, −9.693087002712517827500556474135, −9.689740761619540164774504859100, −8.623741534730103982392013453465, −8.590080283497644573818837122839, −7.62268065385784687130347009087, −7.59210248236904455504245433857, −6.61295427255116933419089721046, −6.43058540328603291594701217505, −5.74653696538159008365274947856, −5.46564141963226094129868273240, −4.66939025450953572028291757581, −4.42620235153241491476235441802, −3.50254127523083272832303269740, −3.36082205887106829291401927249, −1.83373653198188189787556789104, −1.74543966415868381459513149715, −0.796533334320306076603784093106, −0.72311685132929015144546884034,
0.72311685132929015144546884034, 0.796533334320306076603784093106, 1.74543966415868381459513149715, 1.83373653198188189787556789104, 3.36082205887106829291401927249, 3.50254127523083272832303269740, 4.42620235153241491476235441802, 4.66939025450953572028291757581, 5.46564141963226094129868273240, 5.74653696538159008365274947856, 6.43058540328603291594701217505, 6.61295427255116933419089721046, 7.59210248236904455504245433857, 7.62268065385784687130347009087, 8.590080283497644573818837122839, 8.623741534730103982392013453465, 9.689740761619540164774504859100, 9.693087002712517827500556474135, 10.54608039260938214461443661570, 10.63838633763956887465697936871
