| L(s) = 1 | − 14·7-s − 48·9-s + 48·11-s + 28·13-s + 28·17-s − 88·19-s − 28·23-s − 140·29-s + 104·31-s + 364·37-s + 180·41-s + 392·43-s + 56·47-s + 147·49-s + 924·53-s + 568·59-s − 468·61-s + 672·63-s + 224·67-s − 972·71-s + 1.31e3·73-s − 672·77-s + 964·79-s + 1.57e3·81-s + 672·83-s − 1.65e3·89-s − 392·91-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.77·9-s + 1.31·11-s + 0.597·13-s + 0.399·17-s − 1.06·19-s − 0.253·23-s − 0.896·29-s + 0.602·31-s + 1.61·37-s + 0.685·41-s + 1.39·43-s + 0.173·47-s + 3/7·49-s + 2.39·53-s + 1.25·59-s − 0.982·61-s + 1.34·63-s + 0.408·67-s − 1.62·71-s + 2.10·73-s − 0.994·77-s + 1.37·79-s + 2.16·81-s + 0.888·83-s − 1.96·89-s − 0.451·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.562443816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562443816\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 16 p T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 48 T + 2062 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 28 T + 1416 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 28 T + 6566 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 88 T + 15360 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 28 T + 24506 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 140 T + 52502 T^{2} + 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 104 T + 61110 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 364 T + 120606 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 180 T + 3646 T^{2} - 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 392 T + 144414 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 56 T + 179030 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 924 T + 496198 T^{2} - 924 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 568 T + 335888 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 468 T + 353192 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 224 T + 116406 T^{2} - 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 972 T + 950842 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1316 T + 1153374 T^{2} - 1316 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 964 T + 483402 T^{2} - 964 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 672 T + 762256 T^{2} - 672 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1652 T + 1922870 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1428 T + 2332742 T^{2} - 1428 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.14690380272769903734412310835, −9.878476075799528725119323553421, −9.269297340248471613425645332889, −8.888353985389832936604863886299, −8.719077325368560915412978530928, −8.290177435948289459891489228221, −7.46642677453915107889095411878, −7.42950945932524763171549843435, −6.38157856652778788813751105292, −6.33336649154832467263341022022, −5.87985077051909231196385910517, −5.59547321506238545669936287696, −4.71180909429618519957168465504, −4.19631100585705602211265960483, −3.56915392624490267661698196495, −3.37891067844283899104707359869, −2.32244297973380705804394497343, −2.29476537170809633670083383789, −0.906490646171834012620050779679, −0.58064542603283174191661162934,
0.58064542603283174191661162934, 0.906490646171834012620050779679, 2.29476537170809633670083383789, 2.32244297973380705804394497343, 3.37891067844283899104707359869, 3.56915392624490267661698196495, 4.19631100585705602211265960483, 4.71180909429618519957168465504, 5.59547321506238545669936287696, 5.87985077051909231196385910517, 6.33336649154832467263341022022, 6.38157856652778788813751105292, 7.42950945932524763171549843435, 7.46642677453915107889095411878, 8.290177435948289459891489228221, 8.719077325368560915412978530928, 8.888353985389832936604863886299, 9.269297340248471613425645332889, 9.878476075799528725119323553421, 10.14690380272769903734412310835
