| L(s) = 1 | + 3·2-s − 8·3-s + 4-s + 6·5-s − 24·6-s + 28·7-s − 21·8-s + 37·9-s + 18·10-s − 8·12-s + 58·13-s + 84·14-s − 48·15-s − 71·16-s − 17·17-s + 111·18-s − 116·19-s + 6·20-s − 224·21-s − 60·23-s + 168·24-s − 89·25-s + 174·26-s − 80·27-s + 28·28-s − 30·29-s − 144·30-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1.53·3-s + 1/8·4-s + 0.536·5-s − 1.63·6-s + 1.51·7-s − 0.928·8-s + 1.37·9-s + 0.569·10-s − 0.192·12-s + 1.23·13-s + 1.60·14-s − 0.826·15-s − 1.10·16-s − 0.242·17-s + 1.45·18-s − 1.40·19-s + 0.0670·20-s − 2.32·21-s − 0.543·23-s + 1.42·24-s − 0.711·25-s + 1.31·26-s − 0.570·27-s + 0.188·28-s − 0.192·29-s − 0.876·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 17 | \( 1 + p T \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 60 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 172 T + p^{3} T^{2} \) |
| 37 | \( 1 + 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 342 T + p^{3} T^{2} \) |
| 43 | \( 1 - 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 288 T + p^{3} T^{2} \) |
| 53 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 484 T + p^{3} T^{2} \) |
| 71 | \( 1 + 708 T + p^{3} T^{2} \) |
| 73 | \( 1 + 362 T + p^{3} T^{2} \) |
| 79 | \( 1 - 484 T + p^{3} T^{2} \) |
| 83 | \( 1 + 756 T + p^{3} T^{2} \) |
| 89 | \( 1 + 774 T + p^{3} T^{2} \) |
| 97 | \( 1 + 382 T + p^{3} T^{2} \) |
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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
−8.436651985655098341631087818405, −7.32398886975979948715588764577, −6.22490366041311547826621142146, −5.84812157657884637755659701007, −5.29709234944600011466801939855, −4.36666410483750531078792716923, −4.00253625435570803617748140828, −2.26699076022193269076634215280, −1.28243123447466583908947483021, 0,
1.28243123447466583908947483021, 2.26699076022193269076634215280, 4.00253625435570803617748140828, 4.36666410483750531078792716923, 5.29709234944600011466801939855, 5.84812157657884637755659701007, 6.22490366041311547826621142146, 7.32398886975979948715588764577, 8.436651985655098341631087818405
