| L(s) = 1 | + 8·5-s + 10·13-s + 16·17-s + 39·25-s − 40·29-s + 70·37-s − 80·41-s + 49·49-s + 56·53-s + 22·61-s + 80·65-s + 110·73-s + 128·85-s + 160·89-s − 130·97-s − 40·101-s − 182·109-s − 224·113-s + ⋯ |
| L(s) = 1 | + 8/5·5-s + 0.769·13-s + 0.941·17-s + 1.55·25-s − 1.37·29-s + 1.89·37-s − 1.95·41-s + 49-s + 1.05·53-s + 0.360·61-s + 1.23·65-s + 1.50·73-s + 1.50·85-s + 1.79·89-s − 1.34·97-s − 0.396·101-s − 1.66·109-s − 1.98·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.526977928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.526977928\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{2} T^{2} \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 10 T + p^{2} T^{2} \) |
| 17 | \( 1 - 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 40 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 70 T + p^{2} T^{2} \) |
| 41 | \( 1 + 80 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 - 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 160 T + p^{2} T^{2} \) |
| 97 | \( 1 + 130 T + p^{2} T^{2} \) |
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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
−10.35662430577431868338505179069, −9.680694201042961202167672155935, −8.965550391680154665799488317946, −7.904523949423743499712643620828, −6.71657921466068649530501536808, −5.86380711965561486322716731347, −5.23051108084831153486045570307, −3.72466989013489869440663708525, −2.40265884343944723895058450400, −1.26094887772972580207168534305,
1.26094887772972580207168534305, 2.40265884343944723895058450400, 3.72466989013489869440663708525, 5.23051108084831153486045570307, 5.86380711965561486322716731347, 6.71657921466068649530501536808, 7.904523949423743499712643620828, 8.965550391680154665799488317946, 9.680694201042961202167672155935, 10.35662430577431868338505179069
