| L(s) = 1 | + 26·3-s + 50·5-s − 98·7-s + 169·9-s − 14·11-s + 314·13-s + 1.30e3·15-s − 146·17-s + 2.84e3·19-s − 2.54e3·21-s + 3.99e3·23-s + 1.87e3·25-s − 2.47e3·27-s + 6.00e3·29-s + 1.22e3·31-s − 364·33-s − 4.90e3·35-s + 7.23e3·37-s + 8.16e3·39-s + 6.51e3·41-s − 2.33e3·43-s + 8.45e3·45-s + 1.29e4·47-s + 7.20e3·49-s − 3.79e3·51-s + 3.21e3·53-s − 700·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 0.894·5-s − 0.755·7-s + 0.695·9-s − 0.0348·11-s + 0.515·13-s + 1.49·15-s − 0.122·17-s + 1.80·19-s − 1.26·21-s + 1.57·23-s + 3/5·25-s − 0.652·27-s + 1.32·29-s + 0.228·31-s − 0.0581·33-s − 0.676·35-s + 0.868·37-s + 0.859·39-s + 0.605·41-s − 0.192·43-s + 0.622·45-s + 0.853·47-s + 3/7·49-s − 0.204·51-s + 0.157·53-s − 0.0312·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(8.198756964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.198756964\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 26 T + 169 p T^{2} - 26 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 14 T + 250519 T^{2} + 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 314 T + 763535 T^{2} - 314 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 146 T + 2779775 T^{2} + 146 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2848 T + 6233906 T^{2} - 2848 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3992 T + 16110634 T^{2} - 3992 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6006 T + 48676339 T^{2} - 6006 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 1220 T + 57629070 T^{2} - 1220 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7232 T + 51030870 T^{2} - 7232 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6516 T + 187075458 T^{2} - 6516 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2336 T + 257037122 T^{2} + 2336 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12930 T + 158096971 T^{2} - 12930 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3216 T + 614338698 T^{2} - 3216 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6728 T + 77005286 T^{2} - 6728 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 756 T + 1000743434 T^{2} + 756 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 11744 T + 1728084326 T^{2} - 11744 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1904 T + 3475232014 T^{2} + 1904 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 22612 T + 3308577254 T^{2} - 22612 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 62486 T + 5597620295 T^{2} - 62486 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15200 T + 7904251574 T^{2} - 15200 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 42844 T + 10290577282 T^{2} - 42844 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 82822 T + 18061754135 T^{2} - 82822 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.99738882009035943477052917322, −10.81626938250773577290572222031, −10.03708500056612589960234159096, −9.661859022266648393372242455941, −9.164841206354382788615851662756, −9.131928551520238797972517212880, −8.295691267544376732587454430492, −8.197557362360306927329777445057, −7.25056847154449097280560911387, −7.07884213515007263869412585664, −6.23163925891571203453909847899, −5.87575051475195135372233239742, −5.14252117749686473961131630270, −4.59987628050868114513223644318, −3.45349872672106426775438048305, −3.39440550236445549284780588541, −2.52569653463138567642743720399, −2.47384605943586797534128453828, −1.21989225431456724424030529644, −0.78387018593393838207643430373,
0.78387018593393838207643430373, 1.21989225431456724424030529644, 2.47384605943586797534128453828, 2.52569653463138567642743720399, 3.39440550236445549284780588541, 3.45349872672106426775438048305, 4.59987628050868114513223644318, 5.14252117749686473961131630270, 5.87575051475195135372233239742, 6.23163925891571203453909847899, 7.07884213515007263869412585664, 7.25056847154449097280560911387, 8.197557362360306927329777445057, 8.295691267544376732587454430492, 9.131928551520238797972517212880, 9.164841206354382788615851662756, 9.661859022266648393372242455941, 10.03708500056612589960234159096, 10.81626938250773577290572222031, 10.99738882009035943477052917322
