| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 4·7-s + 8-s + 9-s + 10-s + 12-s + 4·14-s + 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s + 4·21-s + 24-s + 25-s + 27-s + 4·28-s − 6·29-s + 30-s − 8·31-s + 32-s + 6·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.755·28-s − 1.11·29-s + 0.182·30-s − 1.43·31-s + 0.176·32-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.577979071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.577979071\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.889742230800007366384340287619, −7.70424926324545431494851724983, −6.90106282892871459571015329925, −5.66012343286953819079657082523, −5.41509779315100847068653879171, −4.57317770756600442017159530038, −3.70681432856719735440542242113, −2.95923023255871163375029903372, −1.87988664142882943456143061246, −1.32801143307487185867201245142,
1.32801143307487185867201245142, 1.87988664142882943456143061246, 2.95923023255871163375029903372, 3.70681432856719735440542242113, 4.57317770756600442017159530038, 5.41509779315100847068653879171, 5.66012343286953819079657082523, 6.90106282892871459571015329925, 7.70424926324545431494851724983, 7.889742230800007366384340287619
