L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s + 6·11-s − 12-s + 13-s − 14-s + 16-s − 3·17-s + 18-s − 4·19-s + 21-s + 6·22-s + 3·23-s − 24-s + 26-s − 27-s − 28-s + 3·29-s + 5·31-s + 32-s − 6·33-s − 3·34-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 + 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.917·19-s + 0.218·21-s + 1.27·22-s + 0.625·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s − 1.04·33-s − 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.211674072\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.211674072\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 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
−9.956655079128140783296437241248, −9.192133894448489417476541893502, −8.258055788314094757493169700102, −6.90032523481236145528972212532, −6.50898555784720230626101441291, −5.76677368707463558412173459318, −4.44782037901496810618606093255, −4.01357092883166224693523319926, −2.63801581631922869292646866349, −1.15770696146574277177343709119,
1.15770696146574277177343709119, 2.63801581631922869292646866349, 4.01357092883166224693523319926, 4.44782037901496810618606093255, 5.76677368707463558412173459318, 6.50898555784720230626101441291, 6.90032523481236145528972212532, 8.258055788314094757493169700102, 9.192133894448489417476541893502, 9.956655079128140783296437241248