| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s + 4·7-s − 4·8-s + 3·9-s + 4·10-s + 6·12-s + 2·13-s − 8·14-s − 4·15-s + 5·16-s − 6·18-s + 14·19-s − 6·20-s + 8·21-s − 10·23-s − 8·24-s + 3·25-s − 4·26-s + 4·27-s + 12·28-s + 10·29-s + 8·30-s + 2·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.51·7-s − 1.41·8-s + 9-s + 1.26·10-s + 1.73·12-s + 0.554·13-s − 2.13·14-s − 1.03·15-s + 5/4·16-s − 1.41·18-s + 3.21·19-s − 1.34·20-s + 1.74·21-s − 2.08·23-s − 1.63·24-s + 3/5·25-s − 0.784·26-s + 0.769·27-s + 2.26·28-s + 1.85·29-s + 1.46·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13176900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.255125473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.255125473\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 68 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 104 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 52 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T - 16 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 131 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 26 T + 323 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 14 T + 240 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.426432853754242984880277711259, −8.290880940125742480169194342953, −8.024458608735070932512844260838, −7.940862215745849590257436088385, −7.44330304873456599546957259680, −7.29661681183445961991519679853, −6.60762056841863508804697227334, −6.49211100950202013893424228844, −5.69219356766473759082316869917, −5.42571212220391165716797986772, −4.74083250715729700734255420967, −4.65103116710337030500424138165, −3.84816005652483532754816466871, −3.59749806580967497483605375454, −3.08368739579778775462255261820, −2.77367095334957668367721959856, −1.93519020991660845067417571667, −1.82625792256567554795106703959, −0.896748755987244239299832428147, −0.877587963729738438618425742141,
0.877587963729738438618425742141, 0.896748755987244239299832428147, 1.82625792256567554795106703959, 1.93519020991660845067417571667, 2.77367095334957668367721959856, 3.08368739579778775462255261820, 3.59749806580967497483605375454, 3.84816005652483532754816466871, 4.65103116710337030500424138165, 4.74083250715729700734255420967, 5.42571212220391165716797986772, 5.69219356766473759082316869917, 6.49211100950202013893424228844, 6.60762056841863508804697227334, 7.29661681183445961991519679853, 7.44330304873456599546957259680, 7.940862215745849590257436088385, 8.024458608735070932512844260838, 8.290880940125742480169194342953, 8.426432853754242984880277711259
