| L(s) = 1 | − 9.70e4·3-s − 3.39e7·5-s − 5.28e8·7-s − 1.03e9·9-s − 1.21e11·11-s + 4.33e11·13-s + 3.29e12·15-s − 1.31e13·17-s + 2.18e13·19-s + 5.13e13·21-s + 1.40e14·23-s + 6.74e14·25-s + 1.11e15·27-s + 1.17e15·29-s + 9.57e14·31-s + 1.18e16·33-s + 1.79e16·35-s − 3.53e16·37-s − 4.20e16·39-s − 1.71e17·41-s + 1.35e17·43-s + 3.51e16·45-s − 5.75e17·47-s − 2.78e17·49-s + 1.27e18·51-s − 9.62e17·53-s + 4.13e18·55-s + ⋯ |
| L(s) = 1 | − 0.949·3-s − 1.55·5-s − 0.707·7-s − 0.0989·9-s − 1.41·11-s + 0.872·13-s + 1.47·15-s − 1.58·17-s + 0.818·19-s + 0.671·21-s + 0.708·23-s + 1.41·25-s + 1.04·27-s + 0.516·29-s + 0.209·31-s + 1.34·33-s + 1.09·35-s − 1.20·37-s − 0.828·39-s − 1.99·41-s + 0.957·43-s + 0.153·45-s − 1.59·47-s − 0.499·49-s + 1.50·51-s − 0.755·53-s + 2.20·55-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
| \(L(11)\) |
\(\approx\) |
\(0.3144680397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3144680397\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 9.70e4T + 1.04e10T^{2} \) |
| 5 | \( 1 + 3.39e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 5.28e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 1.21e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.33e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 1.31e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.18e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.40e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.17e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 9.57e14T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.53e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.71e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.35e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.75e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 9.62e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 4.87e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 4.59e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.78e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.93e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 8.32e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 7.05e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.11e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.62e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 3.84e20T + 5.27e41T^{2} \) |
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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
−16.13289687806881419688853998823, −15.58216471637896654917208473543, −13.13738792238965131739212931221, −11.70900180027629820025749725474, −10.73771173456312031815644336040, −8.386562353775762350214872292221, −6.74928816848479097667158136868, −4.96797949516263459019492487193, −3.23777854840208600449310145602, −0.37457961630098957177918272610,
0.37457961630098957177918272610, 3.23777854840208600449310145602, 4.96797949516263459019492487193, 6.74928816848479097667158136868, 8.386562353775762350214872292221, 10.73771173456312031815644336040, 11.70900180027629820025749725474, 13.13738792238965131739212931221, 15.58216471637896654917208473543, 16.13289687806881419688853998823
