| L(s) = 1 | + 1.83e3·3-s + 3.99e3·5-s + 4.33e5·7-s + 1.77e6·9-s − 1.61e6·11-s − 1.08e7·13-s + 7.32e6·15-s + 6.05e7·17-s + 2.43e8·19-s + 7.95e8·21-s + 6.06e8·23-s − 1.20e9·25-s + 3.34e8·27-s + 5.25e9·29-s + 1.82e9·31-s − 2.97e9·33-s + 1.72e9·35-s − 3.00e9·37-s − 1.99e10·39-s − 4.97e10·41-s − 5.87e10·43-s + 7.08e9·45-s + 4.20e10·47-s + 9.09e10·49-s + 1.11e11·51-s − 1.81e11·53-s − 6.46e9·55-s + ⋯ |
| L(s) = 1 | + 1.45·3-s + 0.114·5-s + 1.39·7-s + 1.11·9-s − 0.275·11-s − 0.625·13-s + 0.166·15-s + 0.608·17-s + 1.18·19-s + 2.02·21-s + 0.853·23-s − 0.986·25-s + 0.166·27-s + 1.64·29-s + 0.369·31-s − 0.400·33-s + 0.159·35-s − 0.192·37-s − 0.908·39-s − 1.63·41-s − 1.41·43-s + 0.127·45-s + 0.569·47-s + 0.938·49-s + 0.884·51-s − 1.12·53-s − 0.0314·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(3.471872560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.471872560\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 68 p^{3} T + p^{13} T^{2} \) |
| 5 | \( 1 - 798 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 433432 T + p^{13} T^{2} \) |
| 11 | \( 1 + 147252 p T + p^{13} T^{2} \) |
| 13 | \( 1 + 10878466 T + p^{13} T^{2} \) |
| 17 | \( 1 - 60569298 T + p^{13} T^{2} \) |
| 19 | \( 1 - 243131740 T + p^{13} T^{2} \) |
| 23 | \( 1 - 606096456 T + p^{13} T^{2} \) |
| 29 | \( 1 - 181332390 p T + p^{13} T^{2} \) |
| 31 | \( 1 - 1824312928 T + p^{13} T^{2} \) |
| 37 | \( 1 + 3005875402 T + p^{13} T^{2} \) |
| 41 | \( 1 + 49704880758 T + p^{13} T^{2} \) |
| 43 | \( 1 + 58766693084 T + p^{13} T^{2} \) |
| 47 | \( 1 - 42095878032 T + p^{13} T^{2} \) |
| 53 | \( 1 + 181140755706 T + p^{13} T^{2} \) |
| 59 | \( 1 + 206730587820 T + p^{13} T^{2} \) |
| 61 | \( 1 + 124479015058 T + p^{13} T^{2} \) |
| 67 | \( 1 + 95665133588 T + p^{13} T^{2} \) |
| 71 | \( 1 - 371436487128 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1800576064726 T + p^{13} T^{2} \) |
| 79 | \( 1 + 1557932091920 T + p^{13} T^{2} \) |
| 83 | \( 1 + 2492790917604 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2994235754490 T + p^{13} T^{2} \) |
| 97 | \( 1 - 4382492665058 T + p^{13} T^{2} \) |
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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
−15.47692787667244616530681597775, −14.43882822513885157922511485792, −13.65534493998788428421583708719, −11.81433028548040862542192349705, −9.970515283868800473717891376329, −8.494965493389112178621067350087, −7.54734228724486521821640464761, −4.91738094808393375782857534035, −3.03462661812754471224571336652, −1.56990393991772904785711319441,
1.56990393991772904785711319441, 3.03462661812754471224571336652, 4.91738094808393375782857534035, 7.54734228724486521821640464761, 8.494965493389112178621067350087, 9.970515283868800473717891376329, 11.81433028548040862542192349705, 13.65534493998788428421583708719, 14.43882822513885157922511485792, 15.47692787667244616530681597775
