| L(s) = 1 | + 15·4-s − 104·11-s + 161·16-s + 40·19-s + 460·29-s − 576·31-s − 244·41-s − 1.56e3·44-s + 110·49-s + 200·59-s + 1.48e3·61-s + 1.45e3·64-s + 656·71-s + 600·76-s + 480·79-s + 660·89-s + 2.43e3·101-s + 1.94e3·109-s + 6.90e3·116-s + 5.45e3·121-s − 8.64e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 2.85·11-s + 2.51·16-s + 0.482·19-s + 2.94·29-s − 3.33·31-s − 0.929·41-s − 5.34·44-s + 0.320·49-s + 0.441·59-s + 3.11·61-s + 2.84·64-s + 1.09·71-s + 0.905·76-s + 0.683·79-s + 0.786·89-s + 2.39·101-s + 1.70·109-s + 5.52·116-s + 4.09·121-s − 6.25·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.115940450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.115940450\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 110 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3910 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9630 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3890 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 288 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 100150 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 122 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 123670 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 142110 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 183510 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 742 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 594470 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 328 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 776590 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 240 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 325370 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 330 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1075390 T^{2} + p^{6} T^{4} \) |
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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
−11.99958183134376279732849961531, −11.28883901914629846712481762418, −11.25571480624891189520173440614, −10.47473042842716163997457062392, −10.24284093269802154477966409202, −10.10114200517281619199281432569, −9.048411507020811325532386508582, −8.294172957580566884743473764199, −8.016594154601279002216180292585, −7.42422128237927050165591689644, −7.11357940921124990242723898276, −6.53740173583214223133516757163, −5.84012799422825265211009764263, −5.18088861471269145153166646197, −5.10895126975731395221963346757, −3.64713644689738041200800205725, −3.05980952932106609294136929095, −2.40800369120816277172154862329, −2.01294238944729982403051570765, −0.68620481326327830775892603488,
0.68620481326327830775892603488, 2.01294238944729982403051570765, 2.40800369120816277172154862329, 3.05980952932106609294136929095, 3.64713644689738041200800205725, 5.10895126975731395221963346757, 5.18088861471269145153166646197, 5.84012799422825265211009764263, 6.53740173583214223133516757163, 7.11357940921124990242723898276, 7.42422128237927050165591689644, 8.016594154601279002216180292585, 8.294172957580566884743473764199, 9.048411507020811325532386508582, 10.10114200517281619199281432569, 10.24284093269802154477966409202, 10.47473042842716163997457062392, 11.25571480624891189520173440614, 11.28883901914629846712481762418, 11.99958183134376279732849961531
