| L(s) = 1 | − 2.26e4·3-s + 9.96e5·5-s − 7.24e7·7-s − 6.49e8·9-s − 4.59e9·11-s + 7.68e9·13-s − 2.25e10·15-s − 6.70e11·17-s + 6.03e11·19-s + 1.64e12·21-s − 1.41e13·23-s − 1.80e13·25-s + 4.10e13·27-s − 1.75e13·29-s + 7.97e13·31-s + 1.04e14·33-s − 7.22e13·35-s − 1.12e15·37-s − 1.73e14·39-s + 3.00e15·41-s + 3.41e15·43-s − 6.47e14·45-s − 1.14e16·47-s − 6.14e15·49-s + 1.51e16·51-s − 2.49e16·53-s − 4.58e15·55-s + ⋯ |
| L(s) = 1 | − 0.664·3-s + 0.228·5-s − 0.678·7-s − 0.559·9-s − 0.587·11-s + 0.201·13-s − 0.151·15-s − 1.37·17-s + 0.429·19-s + 0.450·21-s − 1.63·23-s − 0.947·25-s + 1.03·27-s − 0.224·29-s + 0.541·31-s + 0.390·33-s − 0.154·35-s − 1.41·37-s − 0.133·39-s + 1.43·41-s + 1.03·43-s − 0.127·45-s − 1.49·47-s − 0.539·49-s + 0.910·51-s − 1.03·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.3599809906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3599809906\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 2.26e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 9.96e5T + 1.90e13T^{2} \) |
| 7 | \( 1 + 7.24e7T + 1.13e16T^{2} \) |
| 11 | \( 1 + 4.59e9T + 6.11e19T^{2} \) |
| 13 | \( 1 - 7.68e9T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.70e11T + 2.39e23T^{2} \) |
| 19 | \( 1 - 6.03e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.41e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.75e13T + 6.10e27T^{2} \) |
| 31 | \( 1 - 7.97e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 1.12e15T + 6.24e29T^{2} \) |
| 41 | \( 1 - 3.00e15T + 4.39e30T^{2} \) |
| 43 | \( 1 - 3.41e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.14e16T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.49e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 7.22e15T + 4.42e33T^{2} \) |
| 61 | \( 1 + 1.29e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.74e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.56e17T + 1.49e35T^{2} \) |
| 73 | \( 1 + 8.21e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 2.09e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 3.15e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.78e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 7.58e18T + 5.60e37T^{2} \) |
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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
−11.20677707344193425845425064560, −10.21786085793329570018189068109, −9.105015992117516511514935162328, −7.84657543545554932928712604992, −6.41149534736077177887202918274, −5.76391305178266126426481966333, −4.47454883884177233235317023407, −3.08373219193076662502446918866, −1.90431058885389217353995438086, −0.26192220948299926235573947236,
0.26192220948299926235573947236, 1.90431058885389217353995438086, 3.08373219193076662502446918866, 4.47454883884177233235317023407, 5.76391305178266126426481966333, 6.41149534736077177887202918274, 7.84657543545554932928712604992, 9.105015992117516511514935162328, 10.21786085793329570018189068109, 11.20677707344193425845425064560
