| L(s) = 1 | − 1.44·2-s + 3-s + 0.0867·4-s − 1.93·5-s − 1.44·6-s + 1.10·7-s + 2.76·8-s + 9-s + 2.80·10-s + 6.28·11-s + 0.0867·12-s + 13-s − 1.59·14-s − 1.93·15-s − 4.16·16-s − 6.18·17-s − 1.44·18-s − 0.632·19-s − 0.168·20-s + 1.10·21-s − 9.07·22-s − 8.14·23-s + 2.76·24-s − 1.24·25-s − 1.44·26-s + 27-s + 0.0958·28-s + ⋯ |
| L(s) = 1 | − 1.02·2-s + 0.577·3-s + 0.0433·4-s − 0.867·5-s − 0.589·6-s + 0.417·7-s + 0.977·8-s + 0.333·9-s + 0.885·10-s + 1.89·11-s + 0.0250·12-s + 0.277·13-s − 0.426·14-s − 0.500·15-s − 1.04·16-s − 1.50·17-s − 0.340·18-s − 0.145·19-s − 0.0376·20-s + 0.240·21-s − 1.93·22-s − 1.69·23-s + 0.564·24-s − 0.248·25-s − 0.283·26-s + 0.192·27-s + 0.0181·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 1.44T + 2T^{2} \) |
| 5 | \( 1 + 1.93T + 5T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 11 | \( 1 - 6.28T + 11T^{2} \) |
| 17 | \( 1 + 6.18T + 17T^{2} \) |
| 19 | \( 1 + 0.632T + 19T^{2} \) |
| 23 | \( 1 + 8.14T + 23T^{2} \) |
| 29 | \( 1 - 6.18T + 29T^{2} \) |
| 31 | \( 1 + 6.33T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 + 2.49T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 7.40T + 53T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 + 6.83T + 61T^{2} \) |
| 67 | \( 1 + 5.89T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 + 4.31T + 73T^{2} \) |
| 79 | \( 1 - 9.64T + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 - 7.80T + 89T^{2} \) |
| 97 | \( 1 - 0.781T + 97T^{2} \) |
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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
−8.148098535840720434385621658349, −7.74609374803467002201638970042, −6.78303634641243884194593643159, −6.25446454991763565630876381090, −4.68609485183951457324293934620, −4.17775554706636000569058647998, −3.61457539322090264522602944367, −2.09556507027227510688827975222, −1.34641901536887413276876074931, 0,
1.34641901536887413276876074931, 2.09556507027227510688827975222, 3.61457539322090264522602944367, 4.17775554706636000569058647998, 4.68609485183951457324293934620, 6.25446454991763565630876381090, 6.78303634641243884194593643159, 7.74609374803467002201638970042, 8.148098535840720434385621658349
