| L(s) = 1 | + 4·2-s + 3·3-s + 12·4-s − 10·5-s + 12·6-s − 12·7-s + 32·8-s − 29·9-s − 40·10-s + 12·11-s + 36·12-s − 51·13-s − 48·14-s − 30·15-s + 80·16-s + 66·17-s − 116·18-s + 16·19-s − 120·20-s − 36·21-s + 48·22-s + 96·24-s − 102·25-s − 204·26-s − 120·27-s − 144·28-s − 57·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s − 0.647·7-s + 1.41·8-s − 1.07·9-s − 1.26·10-s + 0.328·11-s + 0.866·12-s − 1.08·13-s − 0.916·14-s − 0.516·15-s + 5/4·16-s + 0.941·17-s − 1.51·18-s + 0.193·19-s − 1.34·20-s − 0.374·21-s + 0.465·22-s + 0.816·24-s − 0.815·25-s − 1.53·26-s − 0.855·27-s − 0.971·28-s − 0.364·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.306353894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.306353894\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - p T + 38 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 p T + 202 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 12 T + 430 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 51 T + 2836 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 66 T + 10258 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 16 T + 6482 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 57 T + 29716 T^{2} + 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 17 T - 27234 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 206 T + 85562 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 373 T + 3836 p T^{2} - 373 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 314 T + 60950 T^{2} + 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 859 T + 380710 T^{2} - 859 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 50 T + 272026 T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 612 T + 438694 T^{2} - 612 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 1062 T + 674530 T^{2} - 1062 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 844 T + 722378 T^{2} - 844 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 399 T + 747574 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1277 T + 1185552 T^{2} + 1277 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 122 T + 977462 T^{2} + 122 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1802 T + 1946542 T^{2} + 1802 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2046 T + 2450554 T^{2} + 2046 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 910 T + 662234 T^{2} - 910 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.622103011906143264368873513862, −9.559732455942510734189188122112, −8.673592385011692908770459838435, −8.467875463698441746435313366816, −8.032903282833144726859367335108, −7.46503660551754130113534982361, −7.08243983359730221808075956664, −7.03898872870363543965092292199, −6.02081332635586191243956338995, −5.94615463452940117249173152937, −5.30291954364579518552346116380, −5.13947473697960181784982474680, −4.12402195399918356516420951915, −4.10281934384568286500270804329, −3.43630114505786450113944420019, −3.19485007537141727028766007010, −2.41622505267621646785079040693, −2.34072548798957623618095641440, −1.23804783775814731569978738582, −0.33125944177404730716119450430,
0.33125944177404730716119450430, 1.23804783775814731569978738582, 2.34072548798957623618095641440, 2.41622505267621646785079040693, 3.19485007537141727028766007010, 3.43630114505786450113944420019, 4.10281934384568286500270804329, 4.12402195399918356516420951915, 5.13947473697960181784982474680, 5.30291954364579518552346116380, 5.94615463452940117249173152937, 6.02081332635586191243956338995, 7.03898872870363543965092292199, 7.08243983359730221808075956664, 7.46503660551754130113534982361, 8.032903282833144726859367335108, 8.467875463698441746435313366816, 8.673592385011692908770459838435, 9.559732455942510734189188122112, 9.622103011906143264368873513862
