L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s − 4·7-s + 8-s + 9-s + 4·10-s − 12-s + 6·13-s − 4·14-s − 4·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 4·20-s + 4·21-s + 4·23-s − 24-s + 11·25-s + 6·26-s − 27-s − 4·28-s − 6·29-s − 4·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s + 1.66·13-s − 1.06·14-s − 1.03·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.894·20-s + 0.872·21-s + 0.834·23-s − 0.204·24-s + 11/5·25-s + 1.17·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.467708208\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.467708208\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 131 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.51441006473715028525396803818, −9.386787693249345075222896043356, −9.078082545589599449667693338271, −7.22075681059514711978726864658, −6.37203718913925795314859712746, −5.94607148812187302229184256149, −5.26387352009640653188048697648, −3.78315375964186311156538524253, −2.77901342456880919028333607166, −1.38783267451649553707956928320,
1.38783267451649553707956928320, 2.77901342456880919028333607166, 3.78315375964186311156538524253, 5.26387352009640653188048697648, 5.94607148812187302229184256149, 6.37203718913925795314859712746, 7.22075681059514711978726864658, 9.078082545589599449667693338271, 9.386787693249345075222896043356, 10.51441006473715028525396803818