| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s + 2·7-s − 4·8-s + 3·9-s + 2·11-s + 6·12-s − 4·13-s − 4·14-s + 5·16-s − 6·17-s − 6·18-s + 8·19-s + 4·21-s − 4·22-s − 2·23-s − 8·24-s + 8·26-s + 4·27-s + 6·28-s + 4·29-s − 6·32-s + 4·33-s + 12·34-s + 9·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 0.603·11-s + 1.73·12-s − 1.10·13-s − 1.06·14-s + 5/4·16-s − 1.45·17-s − 1.41·18-s + 1.83·19-s + 0.872·21-s − 0.852·22-s − 0.417·23-s − 1.63·24-s + 1.56·26-s + 0.769·27-s + 1.13·28-s + 0.742·29-s − 1.06·32-s + 0.696·33-s + 2.05·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.699564518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.699564518\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 130 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.678501056620397074524127119769, −8.646679262043882079660350934941, −7.949294618535340611291231523243, −7.81088795689132586530768703778, −7.34914820501121458376115353228, −7.32297255166330895441215356457, −6.65183061270178398082841946814, −6.54617218469250239751285841744, −5.72116088804945613648435604920, −5.60644202218822576847615714532, −4.81152972952805158930851294096, −4.51933200358358999783950809664, −4.13173782140712780330682031828, −3.55756712798382439185815253945, −2.93856716003623527264808911580, −2.68274395935091927829555649873, −2.12528285019996733619865613515, −1.86071573285890450348112535710, −1.08867540938457529309456509133, −0.65534543893348699897491693365,
0.65534543893348699897491693365, 1.08867540938457529309456509133, 1.86071573285890450348112535710, 2.12528285019996733619865613515, 2.68274395935091927829555649873, 2.93856716003623527264808911580, 3.55756712798382439185815253945, 4.13173782140712780330682031828, 4.51933200358358999783950809664, 4.81152972952805158930851294096, 5.60644202218822576847615714532, 5.72116088804945613648435604920, 6.54617218469250239751285841744, 6.65183061270178398082841946814, 7.32297255166330895441215356457, 7.34914820501121458376115353228, 7.81088795689132586530768703778, 7.949294618535340611291231523243, 8.646679262043882079660350934941, 8.678501056620397074524127119769
