L(s) = 1 | − 0.460·3-s − 0.730·5-s + 3.58·7-s − 2.78·9-s + 3.02·11-s + 0.246·13-s + 0.336·15-s + 17-s − 3.29·19-s − 1.65·21-s − 9.34·23-s − 4.46·25-s + 2.66·27-s − 0.632·29-s + 1.50·31-s − 1.39·33-s − 2.61·35-s − 6.91·37-s − 0.113·39-s − 5.28·41-s + 2.83·43-s + 2.03·45-s + 7.15·47-s + 5.86·49-s − 0.460·51-s − 11.4·53-s − 2.21·55-s + ⋯ |
L(s) = 1 | − 0.266·3-s − 0.326·5-s + 1.35·7-s − 0.929·9-s + 0.912·11-s + 0.0683·13-s + 0.0868·15-s + 0.242·17-s − 0.756·19-s − 0.360·21-s − 1.94·23-s − 0.893·25-s + 0.513·27-s − 0.117·29-s + 0.269·31-s − 0.242·33-s − 0.442·35-s − 1.13·37-s − 0.0181·39-s − 0.824·41-s + 0.431·43-s + 0.303·45-s + 1.04·47-s + 0.838·49-s − 0.0645·51-s − 1.57·53-s − 0.298·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + 0.460T + 3T^{2} \) |
| 5 | \( 1 + 0.730T + 5T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 13 | \( 1 - 0.246T + 13T^{2} \) |
| 19 | \( 1 + 3.29T + 19T^{2} \) |
| 23 | \( 1 + 9.34T + 23T^{2} \) |
| 29 | \( 1 + 0.632T + 29T^{2} \) |
| 31 | \( 1 - 1.50T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 + 5.28T + 41T^{2} \) |
| 43 | \( 1 - 2.83T + 43T^{2} \) |
| 47 | \( 1 - 7.15T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 61 | \( 1 + 2.09T + 61T^{2} \) |
| 67 | \( 1 - 6.17T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 2.44T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−8.219243540316972507714023889861, −7.51067400240005196032285788964, −6.47906576196841501148485868126, −5.86784473181858798892337580132, −5.09680477765861529407751224272, −4.24073108882329944504358216292, −3.62301095943375656703184045199, −2.28914489985431473630714034302, −1.48751109445951892465805851459, 0,
1.48751109445951892465805851459, 2.28914489985431473630714034302, 3.62301095943375656703184045199, 4.24073108882329944504358216292, 5.09680477765861529407751224272, 5.86784473181858798892337580132, 6.47906576196841501148485868126, 7.51067400240005196032285788964, 8.219243540316972507714023889861