| L(s) = 1 | + 40·9-s + 96·11-s + 216·25-s + 392·29-s − 2.28e3·71-s + 2.21e3·79-s − 258·81-s + 3.84e3·99-s − 2.58e3·109-s + 436·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 7.08e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 1.48·9-s + 2.63·11-s + 1.72·25-s + 2.51·29-s − 3.81·71-s + 3.15·79-s − 0.353·81-s + 3.89·99-s − 2.27·109-s + 0.327·121-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 + 3.22·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.469880455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.469880455\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 - 216 T^{2} + p^{6} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 24 T + p^{3} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 3544 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 546 p T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 13252 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 8490 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 98 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 57718 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 41290 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 91242 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 95638 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 147670 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 155158 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 373012 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 207448 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 348022 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 570 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 611434 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 554 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 1128580 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 51438 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 4670 T^{2} + p^{6} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.72689000257927130964560055543, −6.67205068519397142969604947177, −6.35901297086045232665110883546, −6.22582781784321701429339699011, −6.05920442833017574415745601143, −5.52800899471934579558852476002, −5.38118310356915410502638123383, −5.18404901798509013445606818385, −4.64911855751366138853390427147, −4.62960349746326763278507862289, −4.45164522919248861003096956800, −4.25899393711871969241675574884, −3.86750792431483921180251363013, −3.82655269692551834975447049425, −3.42286870557201595723879750195, −3.07818340680556862238828397795, −2.91771730099772911493739243713, −2.59077916588506119237353675036, −2.23676703189619665262037330698, −1.79781906852303569471973259089, −1.44034301181900166392380457009, −1.20871395541835389230433028212, −1.14507777283922984657769050866, −0.811579819976201240688824162733, −0.27474950548510553694643330759,
0.27474950548510553694643330759, 0.811579819976201240688824162733, 1.14507777283922984657769050866, 1.20871395541835389230433028212, 1.44034301181900166392380457009, 1.79781906852303569471973259089, 2.23676703189619665262037330698, 2.59077916588506119237353675036, 2.91771730099772911493739243713, 3.07818340680556862238828397795, 3.42286870557201595723879750195, 3.82655269692551834975447049425, 3.86750792431483921180251363013, 4.25899393711871969241675574884, 4.45164522919248861003096956800, 4.62960349746326763278507862289, 4.64911855751366138853390427147, 5.18404901798509013445606818385, 5.38118310356915410502638123383, 5.52800899471934579558852476002, 6.05920442833017574415745601143, 6.22582781784321701429339699011, 6.35901297086045232665110883546, 6.67205068519397142969604947177, 6.72689000257927130964560055543
