L(s) = 1 | − 2-s − 4-s − 5-s − 7-s + 3·8-s + 10-s + 2·11-s − 13-s + 14-s − 16-s − 4·17-s + 20-s − 2·22-s − 3·23-s + 25-s + 26-s + 28-s − 29-s + 8·31-s − 5·32-s + 4·34-s + 35-s + 4·37-s − 3·40-s + 9·41-s − 8·43-s − 2·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.377·7-s + 1.06·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.188·28-s − 0.185·29-s + 1.43·31-s − 0.883·32-s + 0.685·34-s + 0.169·35-s + 0.657·37-s − 0.474·40-s + 1.40·41-s − 1.21·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5265 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.967833033256388860983506040528, −7.29400997155652741184575997307, −6.53974007232656671800856146389, −5.78723750908105204111581388099, −4.57533895512732778965938982892, −4.32980124235823285316453228548, −3.32464586839880143522492712406, −2.23113386598186817711149388649, −1.05860211671937643906938193219, 0,
1.05860211671937643906938193219, 2.23113386598186817711149388649, 3.32464586839880143522492712406, 4.32980124235823285316453228548, 4.57533895512732778965938982892, 5.78723750908105204111581388099, 6.53974007232656671800856146389, 7.29400997155652741184575997307, 7.967833033256388860983506040528