| L(s) = 1 | − 7.98e3·3-s + 3.09e7·5-s + 9.17e7·7-s − 1.03e10·9-s + 8.72e10·11-s − 2.36e11·13-s − 2.47e11·15-s + 7.42e12·17-s + 4.68e9·19-s − 7.32e11·21-s + 3.33e14·23-s + 4.82e14·25-s + 1.66e14·27-s + 3.23e15·29-s + 6.40e15·31-s − 6.96e14·33-s + 2.84e15·35-s − 1.61e16·37-s + 1.89e15·39-s − 5.77e16·41-s − 2.01e17·43-s − 3.22e17·45-s + 6.62e17·47-s − 5.50e17·49-s − 5.92e16·51-s + 4.62e17·53-s + 2.70e18·55-s + ⋯ |
| L(s) = 1 | − 0.0780·3-s + 1.41·5-s + 0.122·7-s − 0.993·9-s + 1.01·11-s − 0.476·13-s − 0.110·15-s + 0.893·17-s + 0.000175·19-s − 0.00958·21-s + 1.67·23-s + 1.01·25-s + 0.155·27-s + 1.43·29-s + 1.40·31-s − 0.0791·33-s + 0.174·35-s − 0.550·37-s + 0.0371·39-s − 0.671·41-s − 1.42·43-s − 1.40·45-s + 1.83·47-s − 0.984·49-s − 0.0697·51-s + 0.363·53-s + 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.486027196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.486027196\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 7.98e3T + 1.04e10T^{2} \) |
| 5 | \( 1 - 3.09e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 9.17e7T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.72e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.36e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 7.42e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.68e9T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.33e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.23e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.40e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.61e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 5.77e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.01e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 6.62e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 4.62e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 7.39e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.50e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 6.03e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.43e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.48e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 5.70e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.31e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.97e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 9.80e20T + 5.27e41T^{2} \) |
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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
−16.91855361760117672963155216361, −14.66075991442602956618065152803, −13.67593589619787191351571327697, −11.90489149447685943208162220395, −10.13960479246338581818465817597, −8.801034249725583513929749864473, −6.53324167243125699533864651688, −5.20068275431334016419524053049, −2.81308616109381687708896673872, −1.17498568958073832019375028017,
1.17498568958073832019375028017, 2.81308616109381687708896673872, 5.20068275431334016419524053049, 6.53324167243125699533864651688, 8.801034249725583513929749864473, 10.13960479246338581818465817597, 11.90489149447685943208162220395, 13.67593589619787191351571327697, 14.66075991442602956618065152803, 16.91855361760117672963155216361
