L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 4·11-s + 13-s − 15-s + 6·17-s − 4·21-s − 4·23-s + 25-s − 27-s − 6·29-s − 8·31-s − 4·33-s + 4·35-s − 2·37-s − 39-s + 10·41-s − 4·43-s + 45-s + 8·47-s + 9·49-s − 6·51-s − 2·53-s + 4·55-s + 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 1.45·17-s − 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.676·35-s − 0.328·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s − 0.840·51-s − 0.274·53-s + 0.539·55-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.052230097\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.052230097\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.431906608395743092917880999339, −8.692493361509176372458870468102, −7.73395816720361938500916241569, −7.12776183430795663372682016697, −5.86149503664624027165037008263, −5.55234081419335973166658864096, −4.44686046822076618668728454057, −3.64930559196688824116605089224, −1.95119338111066189796333999153, −1.18702392395996184946354056419,
1.18702392395996184946354056419, 1.95119338111066189796333999153, 3.64930559196688824116605089224, 4.44686046822076618668728454057, 5.55234081419335973166658864096, 5.86149503664624027165037008263, 7.12776183430795663372682016697, 7.73395816720361938500916241569, 8.692493361509176372458870468102, 9.431906608395743092917880999339