| L(s) = 1 | + 2·2-s − 5·5-s − 14·7-s − 8·8-s − 10·10-s − 3·11-s − 47·13-s − 28·14-s − 16·16-s − 78·17-s + 64·19-s − 6·22-s + 99·23-s − 94·26-s − 51·29-s − 83·31-s − 156·34-s + 70·35-s + 628·37-s + 128·38-s + 40·40-s + 108·41-s − 299·43-s + 198·46-s − 531·47-s + 343·49-s + 1.12e3·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.0822·11-s − 1.00·13-s − 0.534·14-s − 1/4·16-s − 1.11·17-s + 0.772·19-s − 0.0581·22-s + 0.897·23-s − 0.709·26-s − 0.326·29-s − 0.480·31-s − 0.786·34-s + 0.338·35-s + 2.79·37-s + 0.546·38-s + 0.158·40-s + 0.411·41-s − 1.06·43-s + 0.634·46-s − 1.64·47-s + 49-s + 2.92·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.119952398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119952398\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 p T - 3 p^{2} T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 1322 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 47 T + 12 T^{2} + 47 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 39 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 32 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 99 T - 2366 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 51 T - 21788 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 83 T - 22902 T^{2} + 83 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 314 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 108 T - 57257 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 299 T + 9894 T^{2} + 299 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 531 T + 178138 T^{2} + 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 564 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 12 T - 205235 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 230 T - 174081 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 p T - 51 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 1106 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 739 T + 53082 T^{2} - 739 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1086 T + 607609 T^{2} + 1086 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1642 T + 1783491 T^{2} - 1642 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.764608830913919356627906031424, −9.753326134012964541545393152693, −9.345485164814553059047130600208, −8.886703673602922947762736687169, −8.300244597977675391919936349094, −7.926779977495317475216870007952, −7.40628084820660590996618090734, −6.93898204683007204984349142594, −6.65602663990705078774340325531, −6.15747063606377922517867111551, −5.36858874037411943107665746554, −5.35448569699092403208024011922, −4.48409181020971865935417330692, −4.36897766434579705916657929365, −3.55953139103198219434543786360, −3.29194227301197728120764433060, −2.42291469253189126017682938706, −2.30287016809113354940713719062, −0.986754177133864274022430160216, −0.40843114839121122645625447203,
0.40843114839121122645625447203, 0.986754177133864274022430160216, 2.30287016809113354940713719062, 2.42291469253189126017682938706, 3.29194227301197728120764433060, 3.55953139103198219434543786360, 4.36897766434579705916657929365, 4.48409181020971865935417330692, 5.35448569699092403208024011922, 5.36858874037411943107665746554, 6.15747063606377922517867111551, 6.65602663990705078774340325531, 6.93898204683007204984349142594, 7.40628084820660590996618090734, 7.926779977495317475216870007952, 8.300244597977675391919936349094, 8.886703673602922947762736687169, 9.345485164814553059047130600208, 9.753326134012964541545393152693, 9.764608830913919356627906031424
