| L(s) = 1 | + 2-s − 4-s − 5-s + 4·7-s − 3·8-s − 10-s + 5·11-s + 13-s + 4·14-s − 16-s + 5·19-s + 20-s + 5·22-s − 3·23-s + 25-s + 26-s − 4·28-s − 3·29-s + 3·31-s + 5·32-s − 4·35-s − 5·37-s + 5·38-s + 3·40-s + 2·41-s + 3·43-s − 5·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s − 1.06·8-s − 0.316·10-s + 1.50·11-s + 0.277·13-s + 1.06·14-s − 1/4·16-s + 1.14·19-s + 0.223·20-s + 1.06·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 0.557·29-s + 0.538·31-s + 0.883·32-s − 0.676·35-s − 0.821·37-s + 0.811·38-s + 0.474·40-s + 0.312·41-s + 0.457·43-s − 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.62359357608631, −13.00469591322480, −12.46003433197829, −11.93920543987263, −11.71960686094674, −11.29266618531753, −10.88450671299218, −10.05180066180669, −9.574536629456781, −9.151670520973675, −8.563985671681799, −8.254837684581814, −7.716796052441767, −7.182116204782311, −6.526028042896648, −6.031139910715027, −5.436409608734992, −4.978412173705997, −4.477268978514977, −4.080630923202451, −3.538413919214032, −3.093472215548628, −2.145749434239657, −1.420761964219234, −1.034487206585372, 0,
1.034487206585372, 1.420761964219234, 2.145749434239657, 3.093472215548628, 3.538413919214032, 4.080630923202451, 4.477268978514977, 4.978412173705997, 5.436409608734992, 6.031139910715027, 6.526028042896648, 7.182116204782311, 7.716796052441767, 8.254837684581814, 8.563985671681799, 9.151670520973675, 9.574536629456781, 10.05180066180669, 10.88450671299218, 11.29266618531753, 11.71960686094674, 11.93920543987263, 12.46003433197829, 13.00469591322480, 13.62359357608631
