| L(s) = 1 | + 0.963·2-s + 3-s − 1.07·4-s − 3.45·5-s + 0.963·6-s + 3.91·7-s − 2.95·8-s + 9-s − 3.32·10-s − 3.42·11-s − 1.07·12-s + 2.91·13-s + 3.77·14-s − 3.45·15-s − 0.710·16-s − 17-s + 0.963·18-s + 7.45·19-s + 3.69·20-s + 3.91·21-s − 3.29·22-s − 8.91·23-s − 2.95·24-s + 6.93·25-s + 2.80·26-s + 27-s − 4.19·28-s + ⋯ |
| L(s) = 1 | + 0.681·2-s + 0.577·3-s − 0.535·4-s − 1.54·5-s + 0.393·6-s + 1.47·7-s − 1.04·8-s + 0.333·9-s − 1.05·10-s − 1.03·11-s − 0.309·12-s + 0.807·13-s + 1.00·14-s − 0.891·15-s − 0.177·16-s − 0.242·17-s + 0.227·18-s + 1.70·19-s + 0.827·20-s + 0.853·21-s − 0.703·22-s − 1.85·23-s − 0.604·24-s + 1.38·25-s + 0.550·26-s + 0.192·27-s − 0.792·28-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 | \( 1 - 0.963T + 2T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 - 3.91T + 7T^{2} \) |
| 11 | \( 1 + 3.42T + 11T^{2} \) |
| 13 | \( 1 - 2.91T + 13T^{2} \) |
| 19 | \( 1 - 7.45T + 19T^{2} \) |
| 23 | \( 1 + 8.91T + 23T^{2} \) |
| 29 | \( 1 + 0.411T + 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 + 2.37T + 37T^{2} \) |
| 41 | \( 1 - 7.59T + 41T^{2} \) |
| 43 | \( 1 + 4.15T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 - 5.39T + 53T^{2} \) |
| 59 | \( 1 + 0.0258T + 59T^{2} \) |
| 61 | \( 1 - 5.13T + 61T^{2} \) |
| 67 | \( 1 - 0.256T + 67T^{2} \) |
| 71 | \( 1 - 5.07T + 71T^{2} \) |
| 73 | \( 1 + 4.28T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 8.39T + 83T^{2} \) |
| 89 | \( 1 + 6.35T + 89T^{2} \) |
| 97 | \( 1 + 3.92T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.68992696187635526462902857034, −7.09136871309310452872735974061, −5.67229649861635535013736524091, −5.33639664160392080587212692706, −4.41388467553386887326706326140, −4.01035072728825037085052469113, −3.41025014057071759231805855816, −2.48776973228878793277991940129, −1.26429119359557285852917570546, 0,
1.26429119359557285852917570546, 2.48776973228878793277991940129, 3.41025014057071759231805855816, 4.01035072728825037085052469113, 4.41388467553386887326706326140, 5.33639664160392080587212692706, 5.67229649861635535013736524091, 7.09136871309310452872735974061, 7.68992696187635526462902857034
