| L(s) = 1 | − 1.63·2-s − 3-s + 0.660·4-s + 1.38·5-s + 1.63·6-s + 2.37·7-s + 2.18·8-s + 9-s − 2.26·10-s − 1.95·11-s − 0.660·12-s − 2.02·13-s − 3.87·14-s − 1.38·15-s − 4.88·16-s + 17-s − 1.63·18-s + 2.21·19-s + 0.916·20-s − 2.37·21-s + 3.18·22-s + 3.80·23-s − 2.18·24-s − 3.07·25-s + 3.30·26-s − 27-s + 1.56·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 0.577·3-s + 0.330·4-s + 0.620·5-s + 0.665·6-s + 0.897·7-s + 0.772·8-s + 0.333·9-s − 0.716·10-s − 0.588·11-s − 0.190·12-s − 0.561·13-s − 1.03·14-s − 0.358·15-s − 1.22·16-s + 0.242·17-s − 0.384·18-s + 0.507·19-s + 0.204·20-s − 0.518·21-s + 0.678·22-s + 0.792·23-s − 0.446·24-s − 0.614·25-s + 0.647·26-s − 0.192·27-s + 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 + 2.02T + 13T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 - 3.60T + 43T^{2} \) |
| 47 | \( 1 + 0.0414T + 47T^{2} \) |
| 53 | \( 1 - 2.03T + 53T^{2} \) |
| 59 | \( 1 + 0.596T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 + 4.79T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 9.18T + 89T^{2} \) |
| 97 | \( 1 - 3.30T + 97T^{2} \) |
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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
−7.49666563647926612767823135109, −7.24359763328756405115853206514, −6.00396874379470375103213818534, −5.49477323877711005291339024834, −4.78599920038733208384285167543, −4.12639017326768825331640131932, −2.75070658435938036303342486004, −1.85271081326129195156453688171, −1.15208909939540214838863313864, 0,
1.15208909939540214838863313864, 1.85271081326129195156453688171, 2.75070658435938036303342486004, 4.12639017326768825331640131932, 4.78599920038733208384285167543, 5.49477323877711005291339024834, 6.00396874379470375103213818534, 7.24359763328756405115853206514, 7.49666563647926612767823135109
