| L(s) = 1 | − 81·3-s + 4.03e3·7-s + 6.56e3·9-s + 3.58e4·13-s + 2.58e5·19-s − 3.26e5·21-s + 3.90e5·25-s − 5.31e5·27-s − 1.80e6·31-s − 5.03e5·37-s − 2.90e6·39-s − 3.49e6·43-s + 1.05e7·49-s − 2.09e7·57-s + 2.38e7·61-s + 2.64e7·63-s + 5.42e6·67-s + 1.61e7·73-s − 3.16e7·75-s − 1.88e7·79-s + 4.30e7·81-s + 1.44e8·91-s + 1.46e8·93-s + 1.76e8·97-s + 4.44e7·103-s − 2.03e8·109-s + 4.07e7·111-s + ⋯ |
| L(s) = 1 | − 3-s + 1.68·7-s + 9-s + 1.25·13-s + 1.98·19-s − 1.68·21-s + 25-s − 27-s − 1.95·31-s − 0.268·37-s − 1.25·39-s − 1.02·43-s + 1.82·49-s − 1.98·57-s + 1.72·61-s + 1.68·63-s + 0.269·67-s + 0.569·73-s − 75-s − 0.484·79-s + 81-s + 2.10·91-s + 1.95·93-s + 1.99·97-s + 0.394·103-s − 1.43·109-s + 0.268·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.294795440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294795440\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{4} T \) |
| good | 5 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( 1 - 4034 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 - 35806 T + p^{8} T^{2} \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 - 258526 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 + 1809406 T + p^{8} T^{2} \) |
| 37 | \( 1 + 503522 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 + 3492194 T + p^{8} T^{2} \) |
| 47 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 53 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 59 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 61 | \( 1 - 23826526 T + p^{8} T^{2} \) |
| 67 | \( 1 - 5421406 T + p^{8} T^{2} \) |
| 71 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 73 | \( 1 - 16169282 T + p^{8} T^{2} \) |
| 79 | \( 1 + 18887038 T + p^{8} T^{2} \) |
| 83 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 89 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 97 | \( 1 - 176908034 T + p^{8} T^{2} \) |
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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
−11.28482388997223802256791078736, −10.39223498721639189144440755260, −9.043911697744886252759333097469, −7.912123281311546918097956985348, −6.97214534267064460346636515823, −5.56541434739854537542664646704, −4.98680530962777232607116562015, −3.67859969849509510900533029740, −1.67013562884374873004584515589, −0.906491300799226674814448844546,
0.906491300799226674814448844546, 1.67013562884374873004584515589, 3.67859969849509510900533029740, 4.98680530962777232607116562015, 5.56541434739854537542664646704, 6.97214534267064460346636515823, 7.912123281311546918097956985348, 9.043911697744886252759333097469, 10.39223498721639189144440755260, 11.28482388997223802256791078736
