L(s) = 1 | + 3·3-s − 2·5-s + 6·9-s + 5·11-s + 7·13-s − 6·15-s + 17-s − 2·19-s − 25-s + 9·27-s + 6·29-s + 4·31-s + 15·33-s − 8·37-s + 21·39-s − 2·41-s + 8·43-s − 12·45-s − 10·47-s + 3·51-s − 3·53-s − 10·55-s − 6·57-s − 12·61-s − 14·65-s + 2·67-s + 71-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.894·5-s + 2·9-s + 1.50·11-s + 1.94·13-s − 1.54·15-s + 0.242·17-s − 0.458·19-s − 1/5·25-s + 1.73·27-s + 1.11·29-s + 0.718·31-s + 2.61·33-s − 1.31·37-s + 3.36·39-s − 0.312·41-s + 1.21·43-s − 1.78·45-s − 1.45·47-s + 0.420·51-s − 0.412·53-s − 1.34·55-s − 0.794·57-s − 1.53·61-s − 1.73·65-s + 0.244·67-s + 0.118·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.289127295\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.289127295\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.104202816760128780284259609474, −7.59172695048225035093910717080, −6.57544581355532300151136532127, −6.28439708169582217744643862175, −4.77603057992895308577391448580, −3.90433941011980187080603661274, −3.70390913573535871291062083048, −3.00105416433932316704519838351, −1.80935817096277896864253697299, −1.08835196455667180061053327405,
1.08835196455667180061053327405, 1.80935817096277896864253697299, 3.00105416433932316704519838351, 3.70390913573535871291062083048, 3.90433941011980187080603661274, 4.77603057992895308577391448580, 6.28439708169582217744643862175, 6.57544581355532300151136532127, 7.59172695048225035093910717080, 8.104202816760128780284259609474