| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s + 3·11-s + 12-s − 4·13-s − 4·14-s + 16-s − 3·17-s − 18-s + 4·19-s + 4·21-s − 3·22-s − 24-s + 4·26-s + 27-s + 4·28-s − 4·31-s − 32-s + 3·33-s + 3·34-s + 36-s − 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.917·19-s + 0.872·21-s − 0.639·22-s − 0.204·24-s + 0.784·26-s + 0.192·27-s + 0.755·28-s − 0.718·31-s − 0.176·32-s + 0.522·33-s + 0.514·34-s + 1/6·36-s − 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.647679397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.647679397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−14.12138720348011, −13.82799947496339, −12.94137574197453, −12.32414261307178, −11.98522093824261, −11.44896538511603, −10.95513231953524, −10.58698754067600, −9.753769493184780, −9.479052967630773, −8.877030386957713, −8.535075340540947, −7.888562303041831, −7.460582754185827, −7.095996497947385, −6.467230786154048, −5.663844092938097, −5.039307596250338, −4.598703229683334, −3.931876331972952, −3.229586048951494, −2.492488786764780, −1.827572047677241, −1.498716577523510, −0.5693832837848750,
0.5693832837848750, 1.498716577523510, 1.827572047677241, 2.492488786764780, 3.229586048951494, 3.931876331972952, 4.598703229683334, 5.039307596250338, 5.663844092938097, 6.467230786154048, 7.095996497947385, 7.460582754185827, 7.888562303041831, 8.535075340540947, 8.877030386957713, 9.479052967630773, 9.753769493184780, 10.58698754067600, 10.95513231953524, 11.44896538511603, 11.98522093824261, 12.32414261307178, 12.94137574197453, 13.82799947496339, 14.12138720348011
