| L(s) = 1 | + 81·3-s + 256·4-s − 4.27e3·7-s + 6.56e3·9-s + 2.07e4·12-s + 5.64e4·13-s + 6.55e4·16-s + 1.57e5·19-s − 3.46e5·21-s + 5.31e5·27-s − 1.09e6·28-s + 1.22e6·31-s + 1.67e6·36-s + 5.03e5·37-s + 4.57e6·39-s − 6.83e6·43-s + 5.30e6·48-s + 1.24e7·49-s + 1.44e7·52-s + 1.27e7·57-s − 3.07e5·61-s − 2.80e7·63-s + 1.67e7·64-s − 3.18e7·67-s + 1.61e7·73-s + 4.04e7·76-s − 1.88e7·79-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 1.77·7-s + 9-s + 12-s + 1.97·13-s + 16-s + 1.21·19-s − 1.77·21-s + 27-s − 1.77·28-s + 1.32·31-s + 36-s + 0.268·37-s + 1.97·39-s − 1.99·43-s + 48-s + 2.16·49-s + 1.97·52-s + 1.21·57-s − 0.0222·61-s − 1.77·63-s + 64-s − 1.58·67-s + 0.569·73-s + 1.21·76-s − 0.484·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.486960799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.486960799\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{4} T \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 7 | \( 1 + 4273 T + p^{8} T^{2} \) |
| 11 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 13 | \( 1 - 56447 T + p^{8} T^{2} \) |
| 17 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 19 | \( 1 - 157967 T + p^{8} T^{2} \) |
| 23 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 29 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 31 | \( 1 - 1225967 T + p^{8} T^{2} \) |
| 37 | \( 1 - 503522 T + p^{8} T^{2} \) |
| 41 | \( ( 1 - p^{4} T )( 1 + p^{4} T ) \) |
| 43 | \( 1 + 6837073 T + p^{8} T^{2} \) |
| 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 | \( 1 + 307393 T + p^{8} T^{2} \) |
| 67 | \( 1 + 31874833 T + p^{8} T^{2} \) |
| 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 | \( 1 + 82132513 T + p^{8} T^{2} \) |
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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
−13.10368534062724313031218917458, −11.87076508162835495263786450608, −10.45319324531572178617099602809, −9.532425441090826873081563122439, −8.293713484670541055044943434365, −6.93611703548972549857329832836, −6.10739035442041023086301816218, −3.59241772594880276912088752269, −2.91858936740506823629779242143, −1.22893331001653833683548088976,
1.22893331001653833683548088976, 2.91858936740506823629779242143, 3.59241772594880276912088752269, 6.10739035442041023086301816218, 6.93611703548972549857329832836, 8.293713484670541055044943434365, 9.532425441090826873081563122439, 10.45319324531572178617099602809, 11.87076508162835495263786450608, 13.10368534062724313031218917458
