| L(s) = 1 | − 48·5-s − 238·13-s − 480·17-s + 1.67e3·25-s − 1.68e3·29-s − 2.16e3·37-s − 1.44e3·41-s + 2.40e3·49-s + 5.04e3·53-s + 6.95e3·61-s + 1.14e4·65-s − 1.44e3·73-s + 2.30e4·85-s − 1.24e4·89-s + 1.91e3·97-s + 7.92e3·101-s + 9.36e3·109-s + 6.72e3·113-s + ⋯ |
| L(s) = 1 | − 1.91·5-s − 1.40·13-s − 1.66·17-s + 2.68·25-s − 1.99·29-s − 1.57·37-s − 0.856·41-s + 49-s + 1.79·53-s + 1.86·61-s + 2.70·65-s − 0.270·73-s + 3.18·85-s − 1.57·89-s + 0.203·97-s + 0.776·101-s + 0.787·109-s + 0.526·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4208132362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4208132362\) |
| \(L(3)\) |
|
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 | \( 1 + 48 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 238 T + p^{4} T^{2} \) |
| 17 | \( 1 + 480 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 + 1680 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 2162 T + p^{4} T^{2} \) |
| 41 | \( 1 + 1440 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 - 5040 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 6958 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 1442 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 + 12480 T + p^{4} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{4} T^{2} \) |
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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
−10.26771898585603871509505825547, −9.020200563681054572034505038662, −8.367335571438927412187074366971, −7.23726367205041003174238808864, −7.03634570769983540178640633024, −5.27696740312623297833425327167, −4.35094585938547376606163128172, −3.58341638944473883715560652823, −2.25909301679005787212381276422, −0.32755642714300992490277970814,
0.32755642714300992490277970814, 2.25909301679005787212381276422, 3.58341638944473883715560652823, 4.35094585938547376606163128172, 5.27696740312623297833425327167, 7.03634570769983540178640633024, 7.23726367205041003174238808864, 8.367335571438927412187074366971, 9.020200563681054572034505038662, 10.26771898585603871509505825547
