L(s) = 1 | + 5.90e4·3-s − 3.51e7·5-s + 2.43e8·7-s + 3.48e9·9-s − 5.47e9·11-s − 2.75e11·13-s − 2.07e12·15-s − 7.88e12·17-s − 1.71e13·19-s + 1.43e13·21-s + 1.02e14·23-s + 7.61e14·25-s + 2.05e14·27-s + 3.25e15·29-s − 5.25e15·31-s − 3.23e14·33-s − 8.56e15·35-s − 1.32e15·37-s − 1.62e16·39-s + 1.25e17·41-s − 8.18e16·43-s − 1.22e17·45-s + 3.29e17·47-s − 4.99e17·49-s − 4.65e17·51-s + 1.26e18·53-s + 1.92e17·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.61·5-s + 0.325·7-s + 0.333·9-s − 0.0635·11-s − 0.554·13-s − 0.930·15-s − 0.948·17-s − 0.642·19-s + 0.187·21-s + 0.514·23-s + 1.59·25-s + 0.192·27-s + 1.43·29-s − 1.15·31-s − 0.0367·33-s − 0.524·35-s − 0.0454·37-s − 0.320·39-s + 1.46·41-s − 0.577·43-s − 0.537·45-s + 0.913·47-s − 0.893·49-s − 0.547·51-s + 0.990·53-s + 0.102·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(1.542269376\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542269376\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.90e4T \) |
good | 5 | \( 1 + 3.51e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 2.43e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 5.47e9T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.75e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 7.88e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.71e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.02e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 3.25e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 5.25e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.32e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.25e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.18e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 3.29e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.26e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 5.01e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 7.90e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 9.13e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.27e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.66e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.51e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.78e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.69e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 5.36e20T + 5.27e41T^{2} \) |
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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
−12.94066493649598194601934906117, −11.80347117139358426483154909311, −10.69613369679589813169423232568, −8.919380742769244515081597249875, −7.952461587429409395727375844113, −6.92402663277837857128896666014, −4.73914838199475970225262588449, −3.77067262604501734440173626088, −2.40436483292797185410503477636, −0.62248433116497439205336338380,
0.62248433116497439205336338380, 2.40436483292797185410503477636, 3.77067262604501734440173626088, 4.73914838199475970225262588449, 6.92402663277837857128896666014, 7.952461587429409395727375844113, 8.919380742769244515081597249875, 10.69613369679589813169423232568, 11.80347117139358426483154909311, 12.94066493649598194601934906117