| L(s) = 1 | − 6.56e3·3-s + 6.55e4·4-s − 6.72e6·7-s + 4.30e7·9-s − 4.29e8·12-s − 1.55e9·13-s + 4.29e9·16-s − 9.01e9·19-s + 4.41e10·21-s − 2.82e11·27-s − 4.40e11·28-s − 2.02e11·31-s + 2.82e12·36-s + 6.77e12·37-s + 1.02e13·39-s − 2.33e13·43-s − 2.81e13·48-s + 1.20e13·49-s − 1.01e14·52-s + 5.91e13·57-s − 3.83e14·61-s − 2.89e14·63-s + 2.81e14·64-s − 2.03e14·67-s + 1.35e15·73-s − 5.90e14·76-s − 2.67e15·79-s + ⋯ |
| L(s) = 1 | − 3-s + 4-s − 1.16·7-s + 9-s − 12-s − 1.90·13-s + 16-s − 0.530·19-s + 1.16·21-s − 27-s − 1.16·28-s − 0.237·31-s + 36-s + 1.92·37-s + 1.90·39-s − 1.99·43-s − 48-s + 0.362·49-s − 1.90·52-s + 0.530·57-s − 1.99·61-s − 1.16·63-s + 64-s − 0.502·67-s + 1.67·73-s − 0.530·76-s − 1.76·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{17}{2})\) |
\(\approx\) |
\(1.005034267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.005034267\) |
| \(L(9)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{8} T \) |
| 5 | \( 1 \) |
| good | 2 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 7 | \( 1 + 6728927 T + p^{16} T^{2} \) |
| 11 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 13 | \( 1 + 1554802367 T + p^{16} T^{2} \) |
| 17 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 19 | \( 1 + 9013552993 T + p^{16} T^{2} \) |
| 23 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 29 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 31 | \( 1 + 202786989793 T + p^{16} T^{2} \) |
| 37 | \( 1 - 6771424503358 T + p^{16} T^{2} \) |
| 41 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 43 | \( 1 + 23369166652127 T + p^{16} T^{2} \) |
| 47 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 53 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 59 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 61 | \( 1 + 383320135538113 T + p^{16} T^{2} \) |
| 67 | \( 1 + 203869623664607 T + p^{16} T^{2} \) |
| 71 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 73 | \( 1 - 1351474503392638 T + p^{16} T^{2} \) |
| 79 | \( 1 + 2677497415399678 T + p^{16} T^{2} \) |
| 83 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 89 | \( ( 1 - p^{8} T )( 1 + p^{8} T ) \) |
| 97 | \( 1 - 8929117497058753 T + p^{16} T^{2} \) |
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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
−11.46062364312717689548158301103, −10.28922326114285634954029430163, −9.637839004509038281884726208471, −7.59479817436148322057885274953, −6.75148602939161955905532334587, −5.94744721601054188996887840285, −4.67282333153616724768505820646, −3.10165399627576336474628612671, −1.96549729803649248317144030123, −0.46309872169826666911527805493,
0.46309872169826666911527805493, 1.96549729803649248317144030123, 3.10165399627576336474628612671, 4.67282333153616724768505820646, 5.94744721601054188996887840285, 6.75148602939161955905532334587, 7.59479817436148322057885274953, 9.637839004509038281884726208471, 10.28922326114285634954029430163, 11.46062364312717689548158301103
