| L(s) = 1 | − 28·2-s + 272·4-s − 625·5-s + 2.40e3·7-s + 6.72e3·8-s + 1.75e4·10-s + 2.55e4·11-s − 4.23e4·13-s − 6.72e4·14-s − 3.27e5·16-s + 5.26e5·17-s − 3.50e5·19-s − 1.70e5·20-s − 7.15e5·22-s + 6.21e5·23-s + 3.90e5·25-s + 1.18e6·26-s + 6.53e5·28-s − 6.72e6·29-s − 6.41e6·31-s + 5.72e6·32-s − 1.47e7·34-s − 1.50e6·35-s − 2.31e6·37-s + 9.80e6·38-s − 4.20e6·40-s + 1.02e7·41-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.531·4-s − 0.447·5-s + 0.377·7-s + 0.580·8-s + 0.553·10-s + 0.526·11-s − 0.410·13-s − 0.467·14-s − 1.24·16-s + 1.52·17-s − 0.616·19-s − 0.237·20-s − 0.651·22-s + 0.463·23-s + 1/5·25-s + 0.508·26-s + 0.200·28-s − 1.76·29-s − 1.24·31-s + 0.965·32-s − 1.89·34-s − 0.169·35-s − 0.203·37-s + 0.762·38-s − 0.259·40-s + 0.565·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p^{4} T \) |
| 7 | \( 1 - p^{4} T \) |
| good | 2 | \( 1 + 7 p^{2} T + p^{9} T^{2} \) |
| 11 | \( 1 - 25548 T + p^{9} T^{2} \) |
| 13 | \( 1 + 42306 T + p^{9} T^{2} \) |
| 17 | \( 1 - 526342 T + p^{9} T^{2} \) |
| 19 | \( 1 + 350060 T + p^{9} T^{2} \) |
| 23 | \( 1 - 621976 T + p^{9} T^{2} \) |
| 29 | \( 1 + 6720430 T + p^{9} T^{2} \) |
| 31 | \( 1 + 6412208 T + p^{9} T^{2} \) |
| 37 | \( 1 + 2317682 T + p^{9} T^{2} \) |
| 41 | \( 1 - 10224678 T + p^{9} T^{2} \) |
| 43 | \( 1 - 30114004 T + p^{9} T^{2} \) |
| 47 | \( 1 - 23644912 T + p^{9} T^{2} \) |
| 53 | \( 1 + 57292654 T + p^{9} T^{2} \) |
| 59 | \( 1 + 84934780 T + p^{9} T^{2} \) |
| 61 | \( 1 - 14677822 T + p^{9} T^{2} \) |
| 67 | \( 1 + 244557812 T + p^{9} T^{2} \) |
| 71 | \( 1 + 61901952 T + p^{9} T^{2} \) |
| 73 | \( 1 + 283763726 T + p^{9} T^{2} \) |
| 79 | \( 1 - 276107480 T + p^{9} T^{2} \) |
| 83 | \( 1 - 72995956 T + p^{9} T^{2} \) |
| 89 | \( 1 - 896368470 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1205809578 T + p^{9} T^{2} \) |
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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
−9.404771263085838417905178526998, −8.872427111204542955347259497024, −7.63206021694307665387544518714, −7.41601129980399181310200520151, −5.85965583992689161014928932673, −4.63045087895949411261022765138, −3.53159801677237339550128920219, −1.98156695498797073920329929876, −1.02891164141647243395747791602, 0,
1.02891164141647243395747791602, 1.98156695498797073920329929876, 3.53159801677237339550128920219, 4.63045087895949411261022765138, 5.85965583992689161014928932673, 7.41601129980399181310200520151, 7.63206021694307665387544518714, 8.872427111204542955347259497024, 9.404771263085838417905178526998
