| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 3·11-s − 12-s + 4·13-s − 2·14-s + 16-s + 3·17-s + 18-s − 19-s + 2·21-s − 3·22-s + 6·23-s − 24-s + 4·26-s − 27-s − 2·28-s + 6·29-s − 4·31-s + 32-s + 3·33-s + 3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 0.436·21-s − 0.639·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.568394237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.568394237\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−12.79785805268341, −12.34062947786968, −11.97516020407779, −11.41604221962382, −10.89672561902312, −10.53114503486702, −10.24972077100141, −9.663647372638623, −9.007337384151019, −8.615367216321567, −7.978749169794740, −7.558496622719962, −6.791984889395795, −6.652403616867320, −6.152226160059386, −5.500282875650248, −5.107291833781623, −4.878876405572604, −3.937490977834083, −3.460624132218885, −3.254822648860874, −2.426097900051108, −1.876747964133429, −1.011475190989102, −0.5355870156549921,
0.5355870156549921, 1.011475190989102, 1.876747964133429, 2.426097900051108, 3.254822648860874, 3.460624132218885, 3.937490977834083, 4.878876405572604, 5.107291833781623, 5.500282875650248, 6.152226160059386, 6.652403616867320, 6.791984889395795, 7.558496622719962, 7.978749169794740, 8.615367216321567, 9.007337384151019, 9.663647372638623, 10.24972077100141, 10.53114503486702, 10.89672561902312, 11.41604221962382, 11.97516020407779, 12.34062947786968, 12.79785805268341
