| L(s) = 1 | − 3-s − 5-s + 4·7-s + 9-s − 4·11-s − 13-s + 15-s + 17-s + 8·19-s − 4·21-s + 25-s − 27-s + 2·29-s + 4·33-s − 4·35-s + 10·37-s + 39-s + 10·41-s − 8·43-s − 45-s + 9·49-s − 51-s − 6·53-s + 4·55-s − 8·57-s − 8·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 1.83·19-s − 0.872·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.696·33-s − 0.676·35-s + 1.64·37-s + 0.160·39-s + 1.56·41-s − 1.21·43-s − 0.149·45-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 0.539·55-s − 1.05·57-s − 1.04·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.20343962395182, −12.67629499804607, −12.15449730435773, −11.85053501860245, −11.28451335379664, −11.03481376849542, −10.62544134489379, −10.01353203119881, −9.518980425936175, −9.097588812626354, −8.160381339478464, −8.015339212725277, −7.593723317830135, −7.270908361258666, −6.514268891486664, −5.793239441106107, −5.459178688041407, −4.963923559106742, −4.535723680401739, −4.166934949290559, −3.083768397351530, −2.921358176452570, −2.036072206539844, −1.351284881999295, −0.8688672229918962, 0,
0.8688672229918962, 1.351284881999295, 2.036072206539844, 2.921358176452570, 3.083768397351530, 4.166934949290559, 4.535723680401739, 4.963923559106742, 5.459178688041407, 5.793239441106107, 6.514268891486664, 7.270908361258666, 7.593723317830135, 8.015339212725277, 8.160381339478464, 9.097588812626354, 9.518980425936175, 10.01353203119881, 10.62544134489379, 11.03481376849542, 11.28451335379664, 11.85053501860245, 12.15449730435773, 12.67629499804607, 13.20343962395182
