| L(s) = 1 | − 3-s + 3·5-s − 4·7-s + 9-s + 13-s − 3·15-s + 17-s − 19-s + 4·21-s − 9·23-s + 4·25-s − 27-s − 6·29-s − 2·31-s − 12·35-s − 4·37-s − 39-s + 3·41-s − 7·43-s + 3·45-s + 6·47-s + 9·49-s − 51-s − 6·53-s + 57-s − 6·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.774·15-s + 0.242·17-s − 0.229·19-s + 0.872·21-s − 1.87·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 2.02·35-s − 0.657·37-s − 0.160·39-s + 0.468·41-s − 1.06·43-s + 0.447·45-s + 0.875·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 0.132·57-s − 0.781·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3889763255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3889763255\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.53743924459044, −13.39697046521127, −12.76290082858930, −12.43411096896451, −11.92045952126885, −11.31931513840095, −10.53012802133080, −10.36807221356143, −9.826936027342508, −9.383508436249438, −9.123413484378100, −8.302563885917169, −7.670767744144445, −7.004706188033836, −6.521123811065502, −6.078641841805284, −5.694903536803836, −5.383671516625749, −4.378974204143220, −3.909347799789545, −3.213696924431585, −2.630463957249043, −1.829625006709034, −1.434837265567154, −0.1928775532790656,
0.1928775532790656, 1.434837265567154, 1.829625006709034, 2.630463957249043, 3.213696924431585, 3.909347799789545, 4.378974204143220, 5.383671516625749, 5.694903536803836, 6.078641841805284, 6.521123811065502, 7.004706188033836, 7.670767744144445, 8.302563885917169, 9.123413484378100, 9.383508436249438, 9.826936027342508, 10.36807221356143, 10.53012802133080, 11.31931513840095, 11.92045952126885, 12.43411096896451, 12.76290082858930, 13.39697046521127, 13.53743924459044
