| L(s) = 1 | + 2·2-s + 3·4-s + 2·5-s + 7-s + 4·8-s + 4·10-s + 3·11-s − 2·13-s + 2·14-s + 5·16-s + 9·17-s + 19-s + 6·20-s + 6·22-s + 3·23-s + 3·25-s − 4·26-s + 3·28-s + 3·29-s − 2·31-s + 6·32-s + 18·34-s + 2·35-s − 8·37-s + 2·38-s + 8·40-s + 6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.894·5-s + 0.377·7-s + 1.41·8-s + 1.26·10-s + 0.904·11-s − 0.554·13-s + 0.534·14-s + 5/4·16-s + 2.18·17-s + 0.229·19-s + 1.34·20-s + 1.27·22-s + 0.625·23-s + 3/5·25-s − 0.784·26-s + 0.566·28-s + 0.557·29-s − 0.359·31-s + 1.06·32-s + 3.08·34-s + 0.338·35-s − 1.31·37-s + 0.324·38-s + 1.26·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.369245121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.369245121\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9 T + 46 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 150 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 106 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 112 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 11 T + 102 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 178 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 160 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 216 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.31188709361435731863824481489, −10.25993502607387682591666682057, −9.587789180532392215662842182554, −9.505807757364880381251226260403, −8.546210784056090088603607229462, −8.493100491222144544462042972556, −7.51330203920579775460221038287, −7.50631751385539518485989862848, −6.75402875995307720081969051622, −6.57917823756672287546515249020, −5.77119144029561517694334490406, −5.66468991793911865067055217421, −5.07163255207739999097331877792, −4.82638368208128818212495774063, −4.09105703655908150427078980901, −3.55105060229437274340943099659, −3.05421563973318891113570986544, −2.58005442581466609016613027399, −1.55522268437163874609428800554, −1.35239090928621570494829373116,
1.35239090928621570494829373116, 1.55522268437163874609428800554, 2.58005442581466609016613027399, 3.05421563973318891113570986544, 3.55105060229437274340943099659, 4.09105703655908150427078980901, 4.82638368208128818212495774063, 5.07163255207739999097331877792, 5.66468991793911865067055217421, 5.77119144029561517694334490406, 6.57917823756672287546515249020, 6.75402875995307720081969051622, 7.50631751385539518485989862848, 7.51330203920579775460221038287, 8.493100491222144544462042972556, 8.546210784056090088603607229462, 9.505807757364880381251226260403, 9.587789180532392215662842182554, 10.25993502607387682591666682057, 10.31188709361435731863824481489
