| L(s) = 1 | + 8·2-s − 2·3-s + 34·4-s + 10·5-s − 16·6-s + 96·8-s − 19·9-s + 80·10-s − 14·11-s − 68·12-s − 50·13-s − 20·15-s + 196·16-s + 50·17-s − 152·18-s − 36·19-s + 340·20-s − 112·22-s + 244·23-s − 192·24-s + 75·25-s − 400·26-s + 30·27-s − 26·29-s − 160·30-s + 120·31-s + 352·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.384·3-s + 17/4·4-s + 0.894·5-s − 1.08·6-s + 4.24·8-s − 0.703·9-s + 2.52·10-s − 0.383·11-s − 1.63·12-s − 1.06·13-s − 0.344·15-s + 3.06·16-s + 0.713·17-s − 1.99·18-s − 0.434·19-s + 3.80·20-s − 1.08·22-s + 2.21·23-s − 1.63·24-s + 3/5·25-s − 3.01·26-s + 0.213·27-s − 0.166·29-s − 0.973·30-s + 0.695·31-s + 1.94·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(12.26928997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.26928997\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - p^{3} T + 15 p T^{2} - p^{6} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 14 T + 663 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 50 T + 4987 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 50 T + 387 p T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 36 T + 10170 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 244 T + 29970 T^{2} - 244 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 26 T + 47795 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 120 T - 1618 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 564 T + 173630 T^{2} - 564 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 p T + 133986 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 260 T + 166666 T^{2} + 260 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 350 T + 203423 T^{2} - 350 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 56 T + 265770 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 616 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 336 T + 458858 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 152 T + 599110 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 952 T + p^{3} T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 676 T + 655606 T^{2} + 676 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 376 T + 458918 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 216 T + 1417730 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2742 T + 3608187 T^{2} + 2742 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.00744182014384802302848586624, −11.78337554304876739216182815724, −11.07299026007667274256709614166, −10.93307855172238321430650419227, −10.05158467906896623283892956210, −9.775216624722443456069907555346, −9.000028564575324861953296554889, −8.439499799763868921683104019687, −7.43367384218398661833196368313, −7.24239593767487482425960679859, −6.26931502647373891962714963613, −6.12689537594100828896040468733, −5.45678844671800828457157010558, −5.26250989570475182922531702127, −4.58657810812247511058662225949, −4.27316770750334373277901485334, −3.19278770035578903475279401123, −2.79619774893889984447741283676, −2.35078180802157557984254140315, −0.928496654100638927623280253138,
0.928496654100638927623280253138, 2.35078180802157557984254140315, 2.79619774893889984447741283676, 3.19278770035578903475279401123, 4.27316770750334373277901485334, 4.58657810812247511058662225949, 5.26250989570475182922531702127, 5.45678844671800828457157010558, 6.12689537594100828896040468733, 6.26931502647373891962714963613, 7.24239593767487482425960679859, 7.43367384218398661833196368313, 8.439499799763868921683104019687, 9.000028564575324861953296554889, 9.775216624722443456069907555346, 10.05158467906896623283892956210, 10.93307855172238321430650419227, 11.07299026007667274256709614166, 11.78337554304876739216182815724, 12.00744182014384802302848586624
