| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s − 12-s − 3·13-s − 4·14-s − 15-s + 16-s − 18-s − 19-s + 20-s − 4·21-s − 3·23-s + 24-s + 25-s + 3·26-s − 27-s + 4·28-s − 10·29-s + 30-s + 7·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.832·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.872·21-s − 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s + 0.755·28-s − 1.85·29-s + 0.182·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.30295048045989, −13.08531821085317, −12.31952008536365, −11.88828286293000, −11.57268047575649, −11.07167572975770, −10.68923565613020, −10.08627033810488, −9.811106214581472, −9.227301548638843, −8.590076049086226, −8.254295118941402, −7.657444044962545, −7.291702849275994, −6.780796077835275, −6.101902376258291, −5.594815936932990, −5.208423634963219, −4.584169141969803, −4.177661026136175, −3.328093188360458, −2.511290021830633, −1.923458736461769, −1.614642087500731, −0.8058211387150973, 0,
0.8058211387150973, 1.614642087500731, 1.923458736461769, 2.511290021830633, 3.328093188360458, 4.177661026136175, 4.584169141969803, 5.208423634963219, 5.594815936932990, 6.101902376258291, 6.780796077835275, 7.291702849275994, 7.657444044962545, 8.254295118941402, 8.590076049086226, 9.227301548638843, 9.811106214581472, 10.08627033810488, 10.68923565613020, 11.07167572975770, 11.57268047575649, 11.88828286293000, 12.31952008536365, 13.08531821085317, 13.30295048045989
