| L(s) = 1 | − 24·17-s + 76·25-s + 192·41-s + 194·49-s + 196·73-s − 264·89-s + 428·97-s − 600·113-s + 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 670·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 1.41·17-s + 3.03·25-s + 4.68·41-s + 3.95·49-s + 2.68·73-s − 2.96·89-s + 4.41·97-s − 5.30·113-s + 0.958·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 3.96·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.23598460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.23598460\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 335 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 719 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1250 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1726 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2375 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 48 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3266 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 5990 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 5567 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 5303 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 7778 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 49 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 5593 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 13586 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 107 T + p^{2} T^{2} )^{4} \) |
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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
−6.53602609728570712747593987695, −6.28816077400684475032079229916, −5.95208752329055444344984101525, −5.82677101645902832071840530885, −5.56637800696600782595514513315, −5.30252508903341195663936554887, −5.28121531069069437291103261866, −4.97685686849470007496301557249, −4.61599650135679255199978754908, −4.26369006228061743963224115345, −4.17290698651573591866608294349, −4.14123059712185716588148786618, −4.13247335340617096123584991567, −3.44954046010225112505127528949, −3.12417863571975888186224830844, −2.95932528268430346684067756955, −2.86370323806967105917714288126, −2.32923136887219548673528851126, −2.30427890343271921586680628333, −2.13184933974086871220873150328, −1.63739706059492798699868198898, −1.14710340769354465086855308218, −0.78573954417871106432176824626, −0.63726281208250494753165940225, −0.56179982159950396696951562158,
0.56179982159950396696951562158, 0.63726281208250494753165940225, 0.78573954417871106432176824626, 1.14710340769354465086855308218, 1.63739706059492798699868198898, 2.13184933974086871220873150328, 2.30427890343271921586680628333, 2.32923136887219548673528851126, 2.86370323806967105917714288126, 2.95932528268430346684067756955, 3.12417863571975888186224830844, 3.44954046010225112505127528949, 4.13247335340617096123584991567, 4.14123059712185716588148786618, 4.17290698651573591866608294349, 4.26369006228061743963224115345, 4.61599650135679255199978754908, 4.97685686849470007496301557249, 5.28121531069069437291103261866, 5.30252508903341195663936554887, 5.56637800696600782595514513315, 5.82677101645902832071840530885, 5.95208752329055444344984101525, 6.28816077400684475032079229916, 6.53602609728570712747593987695
