| L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s + 13-s + 16-s + 6·17-s − 4·19-s − 4·22-s + 8·23-s + 26-s − 6·29-s + 8·31-s + 32-s + 6·34-s + 10·37-s − 4·38-s − 6·41-s − 4·43-s − 4·44-s + 8·46-s + 52-s − 10·53-s − 6·58-s + 4·59-s + 2·61-s + 8·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s + 0.277·13-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 0.852·22-s + 1.66·23-s + 0.196·26-s − 1.11·29-s + 1.43·31-s + 0.176·32-s + 1.02·34-s + 1.64·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s + 0.138·52-s − 1.37·53-s − 0.787·58-s + 0.520·59-s + 0.256·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.97962559651324, −12.68455063082979, −12.10913468005719, −11.56037051951420, −11.22247187701189, −10.69923520483092, −10.31821801477721, −9.823494403609838, −9.399407228003549, −8.661324323032651, −8.206182665933502, −7.812550371231130, −7.381361927763595, −6.747197437771060, −6.350150503617041, −5.708627420650803, −5.389985903017861, −4.816790158702010, −4.475343293243679, −3.711195728454867, −3.216541238977752, −2.778801300847461, −2.276398834375772, −1.452192871059858, −0.9175798368156341, 0,
0.9175798368156341, 1.452192871059858, 2.276398834375772, 2.778801300847461, 3.216541238977752, 3.711195728454867, 4.475343293243679, 4.816790158702010, 5.389985903017861, 5.708627420650803, 6.350150503617041, 6.747197437771060, 7.381361927763595, 7.812550371231130, 8.206182665933502, 8.661324323032651, 9.399407228003549, 9.823494403609838, 10.31821801477721, 10.69923520483092, 11.22247187701189, 11.56037051951420, 12.10913468005719, 12.68455063082979, 12.97962559651324
