| L(s) = 1 | + 3.79e4·3-s + 4.30e6·5-s − 6.31e8·7-s − 9.01e9·9-s + 1.24e11·11-s + 2.84e11·13-s + 1.63e11·15-s + 1.33e11·17-s + 2.62e13·19-s − 2.39e13·21-s + 1.05e14·23-s − 4.58e14·25-s − 7.39e14·27-s − 9.82e14·29-s − 8.41e15·31-s + 4.74e15·33-s − 2.71e15·35-s + 1.98e16·37-s + 1.08e16·39-s + 2.34e16·41-s + 1.13e17·43-s − 3.88e16·45-s + 5.56e16·47-s − 1.59e17·49-s + 5.05e15·51-s + 6.13e16·53-s + 5.37e17·55-s + ⋯ |
| L(s) = 1 | + 0.371·3-s + 0.197·5-s − 0.844·7-s − 0.862·9-s + 1.45·11-s + 0.573·13-s + 0.0732·15-s + 0.0160·17-s + 0.983·19-s − 0.313·21-s + 0.529·23-s − 0.961·25-s − 0.691·27-s − 0.433·29-s − 1.84·31-s + 0.539·33-s − 0.166·35-s + 0.679·37-s + 0.212·39-s + 0.272·41-s + 0.800·43-s − 0.170·45-s + 0.154·47-s − 0.286·49-s + 0.00594·51-s + 0.0481·53-s + 0.286·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(2.360081469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360081469\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 3.79e4T + 1.04e10T^{2} \) |
| 5 | \( 1 - 4.30e6T + 4.76e14T^{2} \) |
| 7 | \( 1 + 6.31e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.24e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 2.84e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 1.33e11T + 6.90e25T^{2} \) |
| 19 | \( 1 - 2.62e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.05e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 9.82e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 8.41e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.98e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 2.34e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.13e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.56e16T + 1.30e35T^{2} \) |
| 53 | \( 1 - 6.13e16T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.41e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.42e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 9.92e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.95e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 5.93e16T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.15e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.14e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.18e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 1.12e21T + 5.27e41T^{2} \) |
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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
−11.01512397292966179459096678608, −9.438404706992241780659308151159, −9.085198189044444520145982686619, −7.64378353879904079650935905248, −6.42149147105464932870968091396, −5.58166882349692083141702739303, −3.87285981611699756133187958622, −3.17327223665812229537625329124, −1.84883938454497469721989972166, −0.64576747719857643786766679643,
0.64576747719857643786766679643, 1.84883938454497469721989972166, 3.17327223665812229537625329124, 3.87285981611699756133187958622, 5.58166882349692083141702739303, 6.42149147105464932870968091396, 7.64378353879904079650935905248, 9.085198189044444520145982686619, 9.438404706992241780659308151159, 11.01512397292966179459096678608
