L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 4·7-s + 3·8-s + 9-s + 12-s + 6·13-s − 4·14-s − 16-s − 6·17-s − 18-s + 2·19-s − 4·21-s + 4·23-s − 3·24-s − 5·25-s − 6·26-s − 27-s − 4·28-s + 8·29-s + 6·31-s − 5·32-s + 6·34-s − 36-s − 6·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s + 0.288·12-s + 1.66·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.872·21-s + 0.834·23-s − 0.612·24-s − 25-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + 1.48·29-s + 1.07·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s − 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6902788027\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6902788027\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.39265884605229191617978286710, −11.75312513610494084308792737083, −11.05397435276072073992306547016, −10.19203032281563597278023964338, −8.698507358810087437167594322713, −8.261405216068067893047319044471, −6.79205495113443733345358997575, −5.22981396666096478407481629238, −4.23913097251168411110824061673, −1.38812152386608260002359464899,
1.38812152386608260002359464899, 4.23913097251168411110824061673, 5.22981396666096478407481629238, 6.79205495113443733345358997575, 8.261405216068067893047319044471, 8.698507358810087437167594322713, 10.19203032281563597278023964338, 11.05397435276072073992306547016, 11.75312513610494084308792737083, 13.39265884605229191617978286710