L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 3·11-s + 12-s + 14-s + 16-s + 3·17-s − 18-s − 5·19-s − 21-s + 3·22-s + 6·23-s − 24-s − 5·25-s + 27-s − 28-s − 3·29-s + 4·31-s − 32-s − 3·33-s − 3·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.14·19-s − 0.218·21-s + 0.639·22-s + 1.25·23-s − 0.204·24-s − 25-s + 0.192·27-s − 0.188·28-s − 0.557·29-s + 0.718·31-s − 0.176·32-s − 0.522·33-s − 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.406208490\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.406208490\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−7.83095467611020974508244829542, −7.60029926088136184202795252582, −6.63463689574998734159333102268, −6.02475992278673214962736618499, −5.15343036454333438176623552738, −4.25368038843531296058062561964, −3.31540611856632136893138329245, −2.65233704259189696723118438961, −1.84805669095413120709556733988, −0.63592455640384717903331508777,
0.63592455640384717903331508777, 1.84805669095413120709556733988, 2.65233704259189696723118438961, 3.31540611856632136893138329245, 4.25368038843531296058062561964, 5.15343036454333438176623552738, 6.02475992278673214962736618499, 6.63463689574998734159333102268, 7.60029926088136184202795252582, 7.83095467611020974508244829542