L(s) = 1 | + 3·3-s − 5-s + 7-s + 6·9-s + 5·11-s + 3·13-s − 3·15-s − 17-s + 3·21-s − 6·23-s + 25-s + 9·27-s + 9·29-s − 4·31-s + 15·33-s − 35-s − 2·37-s + 9·39-s + 4·41-s − 10·43-s − 6·45-s + 47-s + 49-s − 3·51-s − 4·53-s − 5·55-s − 8·59-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s + 0.832·13-s − 0.774·15-s − 0.242·17-s + 0.654·21-s − 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s − 0.718·31-s + 2.61·33-s − 0.169·35-s − 0.328·37-s + 1.44·39-s + 0.624·41-s − 1.52·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s − 0.549·53-s − 0.674·55-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(7.137176852\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.137176852\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.20756052440270, −12.56364549360010, −12.20698018428279, −11.71568749255261, −11.22053257875014, −10.62419208737966, −10.14740134021438, −9.529643525948023, −9.202688983829530, −8.711937961612493, −8.367834121356051, −7.921486854682930, −7.570577849409258, −6.754287382760926, −6.538319537213181, −5.967899407494349, −4.985293808818216, −4.507095592388256, −3.979787031845819, −3.559857213930383, −3.250255286664474, −2.422994250615706, −1.845811879811160, −1.420565910128915, −0.6751117107744388,
0.6751117107744388, 1.420565910128915, 1.845811879811160, 2.422994250615706, 3.250255286664474, 3.559857213930383, 3.979787031845819, 4.507095592388256, 4.985293808818216, 5.967899407494349, 6.538319537213181, 6.754287382760926, 7.570577849409258, 7.921486854682930, 8.367834121356051, 8.711937961612493, 9.202688983829530, 9.529643525948023, 10.14740134021438, 10.62419208737966, 11.22053257875014, 11.71568749255261, 12.20698018428279, 12.56364549360010, 13.20756052440270