L(s) = 1 | − 3·9-s − 6·13-s + 8·17-s + 4·29-s − 2·37-s + 10·41-s − 7·49-s + 14·53-s − 12·61-s − 16·73-s + 9·81-s − 10·89-s − 8·97-s − 20·101-s − 20·109-s + 16·113-s + 18·117-s + ⋯ |
L(s) = 1 | − 9-s − 1.66·13-s + 1.94·17-s + 0.742·29-s − 0.328·37-s + 1.56·41-s − 49-s + 1.92·53-s − 1.53·61-s − 1.87·73-s + 81-s − 1.05·89-s − 0.812·97-s − 1.99·101-s − 1.91·109-s + 1.50·113-s + 1.66·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.65439923191156272025601402744, −7.14473555825159979633317300338, −6.12996629302168141583867226312, −5.50747642148206340848918452264, −4.95105732336230764748666309509, −4.00876466448291688011441820105, −2.97502285542379453811613298151, −2.56689996206740720789053854014, −1.24050183102253298299416842270, 0,
1.24050183102253298299416842270, 2.56689996206740720789053854014, 2.97502285542379453811613298151, 4.00876466448291688011441820105, 4.95105732336230764748666309509, 5.50747642148206340848918452264, 6.12996629302168141583867226312, 7.14473555825159979633317300338, 7.65439923191156272025601402744