| L(s) = 1 | − 11·4-s − 28·7-s + 22·13-s + 45·16-s − 132·19-s + 308·28-s − 126·31-s + 354·37-s + 272·43-s + 490·49-s − 242·52-s − 1.17e3·61-s − 363·64-s + 38·67-s + 188·73-s + 1.45e3·76-s − 690·79-s − 616·91-s − 1.98e3·97-s + 1.98e3·103-s − 2.55e3·109-s − 1.26e3·112-s − 4.75e3·121-s + 1.38e3·124-s + 127-s + 131-s + 3.69e3·133-s + ⋯ |
| L(s) = 1 | − 1.37·4-s − 1.51·7-s + 0.469·13-s + 0.703·16-s − 1.59·19-s + 2.07·28-s − 0.730·31-s + 1.57·37-s + 0.964·43-s + 10/7·49-s − 0.645·52-s − 2.45·61-s − 0.708·64-s + 0.0692·67-s + 0.301·73-s + 2.19·76-s − 0.982·79-s − 0.709·91-s − 2.07·97-s + 1.89·103-s − 2.24·109-s − 1.06·112-s − 3.57·121-s + 1.00·124-s + 0.000698·127-s + 0.000666·131-s + 2.40·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 2 | $C_2^2 \wr C_2$ | \( 1 + 11 T^{2} + 19 p^{2} T^{4} + 11 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4757 T^{2} + 9131212 T^{4} + 4757 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 11 T + 334 p T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 16103 T^{2} + 110752648 T^{4} + 16103 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 66 T + 762 p T^{2} + 66 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 16580 T^{2} + 227073517 T^{4} + 16580 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 57852 T^{2} + 1698017789 T^{4} + 57852 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 63 T + 42068 T^{2} + 63 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 177 T + 90632 T^{2} - 177 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 93635 T^{2} + 8952796516 T^{4} + 93635 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 136 T + 147517 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 174744 T^{2} + 15146248718 T^{4} + 174744 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 494267 T^{2} + 102952583068 T^{4} + 494267 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 646299 T^{2} + 187556725712 T^{4} + 646299 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 585 T + 372962 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 19 T + 404134 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 397485 T^{2} + 173980338356 T^{4} + 397485 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 94 T + 606202 T^{2} - 94 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 345 T + 1001934 T^{2} + 345 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 1768119 T^{2} + 1429880388068 T^{4} + 1768119 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 1398400 T^{2} + 1454863323486 T^{4} + 1398400 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 990 T + 1951602 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.63696174865574883611972844360, −6.52948058013293642215949434825, −6.20642830598497974048347734189, −6.17110395030828208201658990292, −6.10702880066989189109128303553, −5.60035410689743955010456760449, −5.47363517113385842461577051025, −5.26852909626236097040674487360, −5.14334382571885853409104576655, −4.59197763588451983805407658445, −4.46219546241992902138788079343, −4.33825213797983399165174840191, −4.25495826248723634826451929876, −3.77225699248095646594731869118, −3.75373105645356305031978132027, −3.52074950236082126130273708813, −3.14588794821273491207127706704, −2.79404003475741885108642862281, −2.76650522359898395153222623748, −2.35800383015989326197550826542, −2.17412756180701093862960969476, −1.78746356226343367574356846662, −1.18601180477300205568132783563, −1.07983027148695969248977662005, −1.01983443144480862861872200845, 0, 0, 0, 0,
1.01983443144480862861872200845, 1.07983027148695969248977662005, 1.18601180477300205568132783563, 1.78746356226343367574356846662, 2.17412756180701093862960969476, 2.35800383015989326197550826542, 2.76650522359898395153222623748, 2.79404003475741885108642862281, 3.14588794821273491207127706704, 3.52074950236082126130273708813, 3.75373105645356305031978132027, 3.77225699248095646594731869118, 4.25495826248723634826451929876, 4.33825213797983399165174840191, 4.46219546241992902138788079343, 4.59197763588451983805407658445, 5.14334382571885853409104576655, 5.26852909626236097040674487360, 5.47363517113385842461577051025, 5.60035410689743955010456760449, 6.10702880066989189109128303553, 6.17110395030828208201658990292, 6.20642830598497974048347734189, 6.52948058013293642215949434825, 6.63696174865574883611972844360
