L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 6·11-s + 15-s + 4·19-s − 21-s + 6·23-s + 25-s − 27-s − 6·29-s − 5·31-s + 6·33-s − 35-s + 2·37-s − 11·43-s − 45-s − 6·47-s − 6·49-s + 6·55-s − 4·57-s − 6·59-s − 61-s + 63-s − 11·67-s − 6·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.258·15-s + 0.917·19-s − 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.898·31-s + 1.04·33-s − 0.169·35-s + 0.328·37-s − 1.67·43-s − 0.149·45-s − 0.875·47-s − 6/7·49-s + 0.809·55-s − 0.529·57-s − 0.781·59-s − 0.128·61-s + 0.125·63-s − 1.34·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6855959819\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6855959819\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.78697038010071, −14.44286478061708, −13.46249863860572, −13.12015041196198, −12.87227919610560, −12.07549277049120, −11.55318359362020, −11.12208260378493, −10.69683326138984, −10.16142293794029, −9.531855568276402, −8.967519004047586, −8.164609128194118, −7.799503564867497, −7.289395362144863, −6.784627350745770, −5.890032045983575, −5.352703683404006, −4.960421836573645, −4.463384664072954, −3.360441545893344, −3.109106372228958, −2.101837436915347, −1.362005310001008, −0.3183968249335827,
0.3183968249335827, 1.362005310001008, 2.101837436915347, 3.109106372228958, 3.360441545893344, 4.463384664072954, 4.960421836573645, 5.352703683404006, 5.890032045983575, 6.784627350745770, 7.289395362144863, 7.799503564867497, 8.164609128194118, 8.967519004047586, 9.531855568276402, 10.16142293794029, 10.69683326138984, 11.12208260378493, 11.55318359362020, 12.07549277049120, 12.87227919610560, 13.12015041196198, 13.46249863860572, 14.44286478061708, 14.78697038010071