L(s) = 1 | − 3-s − 5-s − 2·9-s + 3·11-s + 13-s + 15-s + 3·17-s − 2·19-s − 6·23-s + 25-s + 5·27-s − 9·29-s − 8·31-s − 3·33-s − 10·37-s − 39-s + 2·43-s + 2·45-s + 3·47-s − 3·51-s − 3·55-s + 2·57-s − 12·59-s − 8·61-s − 65-s + 8·67-s + 6·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 0.458·19-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 1.67·29-s − 1.43·31-s − 0.522·33-s − 1.64·37-s − 0.160·39-s + 0.304·43-s + 0.298·45-s + 0.437·47-s − 0.420·51-s − 0.404·55-s + 0.264·57-s − 1.56·59-s − 1.02·61-s − 0.124·65-s + 0.977·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−9.508583967617366827088324595816, −8.796877372751552773865430357179, −7.896967859524111771416111170055, −7.00666421055054313159273263629, −6.01984510959178463345123026067, −5.41652951484887900961612326217, −4.13702557269363707764178227582, −3.35587936623394959029150485777, −1.73203110056877278278191340704, 0,
1.73203110056877278278191340704, 3.35587936623394959029150485777, 4.13702557269363707764178227582, 5.41652951484887900961612326217, 6.01984510959178463345123026067, 7.00666421055054313159273263629, 7.896967859524111771416111170055, 8.796877372751552773865430357179, 9.508583967617366827088324595816