L(s) = 1 | − 782·3-s + 4.09e3·4-s + 1.56e4·5-s + 1.17e5·7-s + 8.00e4·9-s + 2.81e6·11-s − 3.20e6·12-s − 1.95e6·13-s − 1.22e7·15-s + 1.67e7·16-s − 4.77e7·17-s + 6.40e7·20-s − 9.20e7·21-s + 2.44e8·25-s + 3.52e8·27-s + 4.81e8·28-s + 9.13e8·29-s − 2.20e9·33-s + 1.83e9·35-s + 3.28e8·36-s + 1.53e9·39-s + 1.15e10·44-s + 1.25e9·45-s + 9.29e9·47-s − 1.31e10·48-s + 1.38e10·49-s + 3.73e10·51-s + ⋯ |
L(s) = 1 | − 1.07·3-s + 4-s + 5-s + 7-s + 0.150·9-s + 1.59·11-s − 1.07·12-s − 0.405·13-s − 1.07·15-s + 16-s − 1.97·17-s + 20-s − 1.07·21-s + 25-s + 0.911·27-s + 28-s + 1.53·29-s − 1.70·33-s + 35-s + 0.150·36-s + 0.435·39-s + 1.59·44-s + 0.150·45-s + 0.862·47-s − 1.07·48-s + 49-s + 2.12·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{13}{2})\) |
\(\approx\) |
\(2.500669605\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.500669605\) |
\(L(7)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - p^{6} T \) |
| 7 | \( 1 - p^{6} T \) |
good | 2 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 3 | \( 1 + 782 T + p^{12} T^{2} \) |
| 11 | \( 1 - 2817362 T + p^{12} T^{2} \) |
| 13 | \( 1 + 1958542 T + p^{12} T^{2} \) |
| 17 | \( 1 + 47706622 T + p^{12} T^{2} \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( 1 - 913676402 T + p^{12} T^{2} \) |
| 31 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( 1 - 9293086658 T + p^{12} T^{2} \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( 1 + 255286231198 T + p^{12} T^{2} \) |
| 73 | \( 1 + 48396356062 T + p^{12} T^{2} \) |
| 79 | \( 1 - 379435754882 T + p^{12} T^{2} \) |
| 83 | \( 1 - 648698638898 T + p^{12} T^{2} \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( 1 + 859766289982 T + p^{12} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.94363813031684817198633046069, −12.18789605923239860099719537705, −11.38375173389835821275917100084, −10.52583469427103103819344969890, −8.856446019422324245621854040214, −6.83209085099345128773132635889, −6.10222296140079749432957094605, −4.70356052383468584982235803830, −2.27465426506998635043147479361, −1.11925552110328706654861211915,
1.11925552110328706654861211915, 2.27465426506998635043147479361, 4.70356052383468584982235803830, 6.10222296140079749432957094605, 6.83209085099345128773132635889, 8.856446019422324245621854040214, 10.52583469427103103819344969890, 11.38375173389835821275917100084, 12.18789605923239860099719537705, 13.94363813031684817198633046069