| L(s) = 1 | − 2-s + 4-s − 8-s − 2·11-s + 2·13-s + 16-s + 2·22-s + 4·23-s − 5·25-s − 2·26-s + 4·29-s − 32-s − 8·37-s + 2·41-s − 2·44-s − 4·46-s + 5·50-s + 2·52-s − 2·53-s − 4·58-s + 4·59-s − 12·61-s + 64-s − 8·67-s + 12·71-s − 14·73-s + 8·74-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s + 0.554·13-s + 1/4·16-s + 0.426·22-s + 0.834·23-s − 25-s − 0.392·26-s + 0.742·29-s − 0.176·32-s − 1.31·37-s + 0.312·41-s − 0.301·44-s − 0.589·46-s + 0.707·50-s + 0.277·52-s − 0.274·53-s − 0.525·58-s + 0.520·59-s − 1.53·61-s + 1/8·64-s − 0.977·67-s + 1.42·71-s − 1.63·73-s + 0.929·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.02344914786311, −12.53823147003844, −12.10068664013152, −11.57399020353862, −11.16517662535643, −10.68993830189432, −10.24401628305302, −9.943647527258047, −9.222939443050164, −8.925571754273007, −8.411950133410062, −7.981843607745507, −7.468536286987948, −7.048739037617469, −6.496307886154064, −5.975849648894616, −5.516872806696208, −4.956603863095136, −4.350501835797325, −3.760917693091957, −3.042468850365920, −2.780450105991890, −1.892583333128569, −1.497416409396638, −0.7058281423132659, 0,
0.7058281423132659, 1.497416409396638, 1.892583333128569, 2.780450105991890, 3.042468850365920, 3.760917693091957, 4.350501835797325, 4.956603863095136, 5.516872806696208, 5.975849648894616, 6.496307886154064, 7.048739037617469, 7.468536286987948, 7.981843607745507, 8.411950133410062, 8.925571754273007, 9.222939443050164, 9.943647527258047, 10.24401628305302, 10.68993830189432, 11.16517662535643, 11.57399020353862, 12.10068664013152, 12.53823147003844, 13.02344914786311
