| L(s) = 1 | − 243·3-s + 1.09e4·7-s + 5.90e4·9-s − 5.60e5·13-s + 4.92e6·19-s − 2.65e6·21-s − 1.43e7·27-s + 5.16e5·31-s + 1.35e8·37-s + 1.36e8·39-s + 7.16e7·43-s − 1.63e8·49-s − 1.19e9·57-s − 1.35e9·61-s + 6.44e8·63-s − 1.26e9·67-s − 4.14e9·73-s + 3.95e9·79-s + 3.48e9·81-s − 6.11e9·91-s − 1.25e8·93-s − 1.52e10·97-s − 3.33e9·103-s + 3.06e10·109-s − 3.28e10·111-s − 3.30e10·117-s + ⋯ |
| L(s) = 1 | − 3-s + 0.648·7-s + 9-s − 1.50·13-s + 1.98·19-s − 0.648·21-s − 27-s + 0.0180·31-s + 1.94·37-s + 1.50·39-s + 0.487·43-s − 0.578·49-s − 1.98·57-s − 1.60·61-s + 0.648·63-s − 0.933·67-s − 1.99·73-s + 1.28·79-s + 81-s − 0.979·91-s − 0.0180·93-s − 1.78·97-s − 0.287·103-s + 1.99·109-s − 1.94·111-s − 1.50·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.532798245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.532798245\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{5} T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 10907 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 13 | \( 1 + 560257 T + p^{10} T^{2} \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( 1 - 4926251 T + p^{10} T^{2} \) |
| 23 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 - 516899 T + p^{10} T^{2} \) |
| 37 | \( 1 - 135214586 T + p^{10} T^{2} \) |
| 41 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 43 | \( 1 - 71672243 T + p^{10} T^{2} \) |
| 47 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 53 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 59 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 61 | \( 1 + 1354266001 T + p^{10} T^{2} \) |
| 67 | \( 1 + 1260882493 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 + 4144040686 T + p^{10} T^{2} \) |
| 79 | \( 1 - 3959005298 T + p^{10} T^{2} \) |
| 83 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 + 15296411593 T + p^{10} T^{2} \) |
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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
−10.00242154838156556725650103815, −9.373626040606567214536110891062, −7.75315459278993474128323931234, −7.27718360820557815045474676943, −5.98940610050213450580203188864, −5.09943947350465398301019515779, −4.41627094952894307285389079839, −2.87776272282012113104507370875, −1.55543835713310921846943537598, −0.57859740311418210128024734699,
0.57859740311418210128024734699, 1.55543835713310921846943537598, 2.87776272282012113104507370875, 4.41627094952894307285389079839, 5.09943947350465398301019515779, 5.98940610050213450580203188864, 7.27718360820557815045474676943, 7.75315459278993474128323931234, 9.373626040606567214536110891062, 10.00242154838156556725650103815
