| L(s) = 1 | − 34·9-s − 100·11-s − 352·23-s − 314·25-s + 260·29-s + 212·37-s − 540·43-s + 16·53-s + 1.94e3·67-s − 2.24e3·71-s + 1.04e3·79-s − 6·81-s + 3.40e3·99-s − 3.76e3·107-s − 516·109-s − 2.31e3·113-s + 2.09e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.55e3·169-s + ⋯ |
| L(s) = 1 | − 1.25·9-s − 2.74·11-s − 3.19·23-s − 2.51·25-s + 1.66·29-s + 0.941·37-s − 1.91·43-s + 0.0414·53-s + 3.54·67-s − 3.75·71-s + 1.49·79-s − 0.00823·81-s + 3.45·99-s − 3.40·107-s − 0.453·109-s − 1.92·113-s + 1.57·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 2.52·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 34 T^{2} + 1162 T^{4} + 34 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 314 T^{2} + 48034 T^{4} + 314 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 50 T + 2702 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5554 T^{2} + 17209282 T^{4} + 5554 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 5088 T^{2} + 11466434 T^{4} + 5088 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 12690 T^{2} + 132581162 T^{4} + 12690 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 176 T + 31038 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 130 T + 49818 T^{2} - 130 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 19060 T^{2} - 9966698 T^{4} + 19060 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 106 T + 103530 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 208848 T^{2} + 19595539298 T^{4} + 208848 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 270 T + 97614 T^{2} + 270 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 228052 T^{2} + 34127693334 T^{4} + 228052 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 276710 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 632162 T^{2} + 176905491658 T^{4} + 632162 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 8978 p T^{2} + 150597445698 T^{4} + 8978 p^{7} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 972 T + 743862 T^{2} - 972 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 1124 T + 1018926 T^{2} + 1124 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 1279312 T^{2} + 705108016354 T^{4} + 1279312 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 524 T + 480382 T^{2} - 524 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 2275682 T^{2} + 1948536513034 T^{4} + 2275682 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 804000 T^{2} + 717141036962 T^{4} - 804000 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 567808 T^{2} + 211724872834 T^{4} + 567808 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−7.67347132308938388712881341662, −7.36244611043670734139846057452, −6.84531822817553192067199005928, −6.72739205262929589871082304925, −6.57921653727897942312079437815, −6.02524052429371110738179663019, −5.91507306124162811808865041530, −5.86649899573915052560316925046, −5.83478179653561167573232596452, −5.19449002473648229206522963413, −5.09342385535846629269960596578, −5.00558760729438865379857265577, −4.75825150322091507481511568258, −4.23856567075930751759195213981, −3.86754442393846248465317843168, −3.84716770434027941673646726498, −3.73464064429804457709493125821, −3.10483405968378178166773960212, −2.74615417524793132459023928891, −2.70859903348470520344603812480, −2.40001673600326467153220223563, −2.06299968549217676939768060095, −2.00135052437211052823829242174, −1.17802198936285649311244524486, −1.16799725564934247889342696382, 0, 0, 0, 0,
1.16799725564934247889342696382, 1.17802198936285649311244524486, 2.00135052437211052823829242174, 2.06299968549217676939768060095, 2.40001673600326467153220223563, 2.70859903348470520344603812480, 2.74615417524793132459023928891, 3.10483405968378178166773960212, 3.73464064429804457709493125821, 3.84716770434027941673646726498, 3.86754442393846248465317843168, 4.23856567075930751759195213981, 4.75825150322091507481511568258, 5.00558760729438865379857265577, 5.09342385535846629269960596578, 5.19449002473648229206522963413, 5.83478179653561167573232596452, 5.86649899573915052560316925046, 5.91507306124162811808865041530, 6.02524052429371110738179663019, 6.57921653727897942312079437815, 6.72739205262929589871082304925, 6.84531822817553192067199005928, 7.36244611043670734139846057452, 7.67347132308938388712881341662
