| L(s) = 1 | + 5.90e4·3-s − 1.13e7·5-s + 8.09e8·7-s + 3.48e9·9-s + 5.57e10·11-s − 2.33e11·13-s − 6.67e11·15-s − 9.51e12·17-s − 1.41e13·19-s + 4.78e13·21-s − 1.44e14·23-s − 3.49e14·25-s + 2.05e14·27-s + 1.75e15·29-s − 1.79e15·31-s + 3.29e15·33-s − 9.15e15·35-s + 5.36e16·37-s − 1.37e16·39-s + 6.45e16·41-s − 1.20e17·43-s − 3.94e16·45-s − 6.12e17·47-s + 9.75e16·49-s − 5.61e17·51-s − 3.70e17·53-s − 6.30e17·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.517·5-s + 1.08·7-s + 0.333·9-s + 0.648·11-s − 0.469·13-s − 0.298·15-s − 1.14·17-s − 0.530·19-s + 0.625·21-s − 0.729·23-s − 0.731·25-s + 0.192·27-s + 0.776·29-s − 0.393·31-s + 0.374·33-s − 0.561·35-s + 1.83·37-s − 0.271·39-s + 0.750·41-s − 0.847·43-s − 0.172·45-s − 1.69·47-s + 0.174·49-s − 0.660·51-s − 0.291·53-s − 0.335·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 1.13e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 8.09e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 5.57e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 2.33e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 9.51e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 1.41e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.44e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.75e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.79e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 5.36e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 6.45e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.20e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 6.12e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 3.70e17T + 1.62e36T^{2} \) |
| 59 | \( 1 - 4.74e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 8.07e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.31e17T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.36e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 2.16e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 2.92e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.33e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 4.98e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 6.53e20T + 5.27e41T^{2} \) |
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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
−11.07082666691586890221115048900, −9.651115396387438328219704939220, −8.465985842189509180471110653448, −7.71489254652025955122157898640, −6.41495698140621031390294932295, −4.72638934636013517668819527317, −3.96650358143931719579826948386, −2.45426700419863481786839618569, −1.46539072560844591562597806557, 0,
1.46539072560844591562597806557, 2.45426700419863481786839618569, 3.96650358143931719579826948386, 4.72638934636013517668819527317, 6.41495698140621031390294932295, 7.71489254652025955122157898640, 8.465985842189509180471110653448, 9.651115396387438328219704939220, 11.07082666691586890221115048900
