| L(s) = 1 | + 1.02e3·4-s + 6.24e3·5-s − 1.68e4·7-s + 5.90e4·9-s + 3.71e5·13-s + 1.04e6·16-s + 2.14e6·19-s + 6.39e6·20-s − 6.46e6·23-s + 2.92e7·25-s − 1.72e7·28-s − 4.72e6·29-s − 4.21e7·31-s − 1.04e8·35-s + 6.04e7·36-s − 2.21e8·41-s − 8.43e7·43-s + 3.68e8·45-s + 4.06e8·47-s + 2.82e8·49-s + 3.80e8·52-s + 5.45e8·53-s − 1.33e9·59-s − 9.92e8·63-s + 1.07e9·64-s + 2.31e9·65-s − 3.48e9·73-s + ⋯ |
| L(s) = 1 | + 4-s + 1.99·5-s − 7-s + 9-s + 13-s + 16-s + 0.867·19-s + 1.99·20-s − 1.00·23-s + 2.99·25-s − 28-s − 0.230·29-s − 1.47·31-s − 1.99·35-s + 36-s − 1.90·41-s − 0.573·43-s + 1.99·45-s + 1.77·47-s + 49-s + 52-s + 1.30·53-s − 1.86·59-s − 63-s + 64-s + 1.99·65-s − 1.68·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(4.280678516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.280678516\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + p^{5} T \) |
| 13 | \( 1 - p^{5} T \) |
| good | 2 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 3 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 5 | \( 1 - 6243 T + p^{10} T^{2} \) |
| 11 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 17 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 19 | \( 1 - 2147375 T + p^{10} T^{2} \) |
| 23 | \( 1 + 6466725 T + p^{10} T^{2} \) |
| 29 | \( 1 + 4721673 T + p^{10} T^{2} \) |
| 31 | \( 1 + 42179225 T + p^{10} T^{2} \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( 1 + 221229150 T + p^{10} T^{2} \) |
| 43 | \( 1 + 84317525 T + p^{10} T^{2} \) |
| 47 | \( 1 - 406710639 T + p^{10} T^{2} \) |
| 53 | \( 1 - 545143575 T + p^{10} T^{2} \) |
| 59 | \( 1 + 1330532550 T + p^{10} T^{2} \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( 1 + 3488411189 T + p^{10} T^{2} \) |
| 79 | \( 1 - 5013092827 T + p^{10} T^{2} \) |
| 83 | \( 1 - 893026839 T + p^{10} T^{2} \) |
| 89 | \( 1 - 3850158675 T + p^{10} T^{2} \) |
| 97 | \( 1 + 9195272389 T + p^{10} T^{2} \) |
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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
−12.23116874392716546508043210527, −10.60882419153118837687103128862, −10.03306962232178709721904959211, −9.097761705280783524899569333665, −7.17717030272262181642486514594, −6.30080664935519594460025193152, −5.55626747476332430665909574137, −3.41928952876231657969455093576, −2.10369014234273439765813355288, −1.27257268861384825727314505015,
1.27257268861384825727314505015, 2.10369014234273439765813355288, 3.41928952876231657969455093576, 5.55626747476332430665909574137, 6.30080664935519594460025193152, 7.17717030272262181642486514594, 9.097761705280783524899569333665, 10.03306962232178709721904959211, 10.60882419153118837687103128862, 12.23116874392716546508043210527
