| L(s) = 1 | + 52·9-s + 250·25-s + 216·29-s − 686·49-s + 570·81-s − 9.06e3·109-s + 5.04e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 5.54e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 1.30e4·225-s + ⋯ |
| L(s) = 1 | + 1.92·9-s + 2·25-s + 1.38·29-s − 2·49-s + 0.781·81-s − 7.96·109-s + 3.78·121-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 + 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 + 0.000326·211-s + 0.000300·223-s + 3.85·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.001443329698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001443329698\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2}( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 2774 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 754 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 175646 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 828 T + p^{3} T^{2} )^{2}( 1 + 828 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 504254 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 236 T + p^{3} T^{2} )^{2}( 1 + 236 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 1141306 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 897874 T^{2} + p^{6} T^{4} )^{2} \) |
| 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
−7.39666497374818726661506415918, −7.09184252326596840360865359561, −6.75286225617605263297537849541, −6.53952725778026332426293031947, −6.53851207909093760361024587151, −6.42663675288941001808518713762, −5.78828494939879517822899422819, −5.66805708792017094345248161760, −5.23888242681101331221276215848, −5.08620032099521976642001963032, −4.76311010377699498829224649719, −4.54004713186749754670027663461, −4.48398956067161688034772253845, −3.97615223445963869253852007286, −3.85024331626436153801437681741, −3.48260769112308358706784146607, −3.12631270814052824390644090198, −2.80457394499368859021453723721, −2.57192885542672481855826070835, −2.23358587901637427027205855716, −1.65927082672215784694885065595, −1.38322762912647628383538295491, −1.09192184039771230722238159706, −0.963663993011681033600297896480, −0.00466066659332570932309269716,
0.00466066659332570932309269716, 0.963663993011681033600297896480, 1.09192184039771230722238159706, 1.38322762912647628383538295491, 1.65927082672215784694885065595, 2.23358587901637427027205855716, 2.57192885542672481855826070835, 2.80457394499368859021453723721, 3.12631270814052824390644090198, 3.48260769112308358706784146607, 3.85024331626436153801437681741, 3.97615223445963869253852007286, 4.48398956067161688034772253845, 4.54004713186749754670027663461, 4.76311010377699498829224649719, 5.08620032099521976642001963032, 5.23888242681101331221276215848, 5.66805708792017094345248161760, 5.78828494939879517822899422819, 6.42663675288941001808518713762, 6.53851207909093760361024587151, 6.53952725778026332426293031947, 6.75286225617605263297537849541, 7.09184252326596840360865359561, 7.39666497374818726661506415918
