L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 9-s + 2·10-s − 11-s + 2·12-s − 5·13-s + 15-s − 4·16-s − 4·17-s + 2·18-s − 2·19-s + 2·20-s − 2·22-s + 3·23-s + 25-s − 10·26-s + 27-s − 3·29-s + 2·30-s − 8·32-s − 33-s − 8·34-s + 2·36-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.38·13-s + 0.258·15-s − 16-s − 0.970·17-s + 0.471·18-s − 0.458·19-s + 0.447·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 1.96·26-s + 0.192·27-s − 0.557·29-s + 0.365·30-s − 1.41·32-s − 0.174·33-s − 1.37·34-s + 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.07011865588971992381805149977, −6.83590993670011109204535185673, −5.95459795783327623224384307503, −5.03435991417994298460060284193, −4.85658883069947606661681695080, −3.92778897931413061912634353290, −3.16751499488447655143976134188, −2.44634072912272836054580876493, −1.86206803884858170120885056801, 0,
1.86206803884858170120885056801, 2.44634072912272836054580876493, 3.16751499488447655143976134188, 3.92778897931413061912634353290, 4.85658883069947606661681695080, 5.03435991417994298460060284193, 5.95459795783327623224384307503, 6.83590993670011109204535185673, 7.07011865588971992381805149977