| L(s) = 1 | + 8·5-s + 36·17-s − 2·25-s − 8·29-s + 144·37-s + 36·41-s + 50·49-s + 88·53-s − 144·61-s + 164·73-s + 288·85-s + 252·89-s + 220·97-s + 184·101-s − 288·109-s + 252·113-s + 194·121-s − 344·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 2.11·17-s − 0.0799·25-s − 0.275·29-s + 3.89·37-s + 0.878·41-s + 1.02·49-s + 1.66·53-s − 2.36·61-s + 2.24·73-s + 3.38·85-s + 2.83·89-s + 2.26·97-s + 1.82·101-s − 2.64·109-s + 2.23·113-s + 1.60·121-s − 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.441·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.584873804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.584873804\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 670 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2690 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8546 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8354 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8594 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3550 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.013018884498517853558631023492, −8.922676147648971169630142531606, −8.083622918615198645903858220743, −7.78619995681552466229885532373, −7.54661473075929990952076346392, −7.31233820175240598183256722868, −6.29821566146144892784002987822, −6.26397869321960292765317028505, −5.81062172813323384564936501985, −5.80813606062610157076197419598, −4.94635786834031838701630724880, −4.93575695541660474797804156057, −4.06410165079592737234686610873, −3.78339677627036024396135804544, −3.18447255239663923278878685409, −2.57153676511643879580590706307, −2.28962900927643314952042269868, −1.75156446775331973202373380216, −0.941770219425696980968359119396, −0.78262324849015727884478115642,
0.78262324849015727884478115642, 0.941770219425696980968359119396, 1.75156446775331973202373380216, 2.28962900927643314952042269868, 2.57153676511643879580590706307, 3.18447255239663923278878685409, 3.78339677627036024396135804544, 4.06410165079592737234686610873, 4.93575695541660474797804156057, 4.94635786834031838701630724880, 5.80813606062610157076197419598, 5.81062172813323384564936501985, 6.26397869321960292765317028505, 6.29821566146144892784002987822, 7.31233820175240598183256722868, 7.54661473075929990952076346392, 7.78619995681552466229885532373, 8.083622918615198645903858220743, 8.922676147648971169630142531606, 9.013018884498517853558631023492
