L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 3·7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s − 13-s − 3·14-s + 15-s + 16-s − 3·17-s + 18-s − 2·19-s + 20-s − 3·21-s − 2·22-s − 5·23-s + 24-s + 25-s − 26-s + 27-s − 3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s − 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 0.654·21-s − 0.426·22-s − 1.04·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.99027345191888, −12.70727710446692, −12.45712709202838, −11.77466411807223, −11.32854403362590, −10.67675682443583, −10.27465325730215, −9.968039537823606, −9.470467024667608, −8.963524014733683, −8.480622617253920, −7.989807736965238, −7.429730091394514, −6.896279086350295, −6.471340044518876, −6.200457980775400, −5.428902477694703, −5.135444729366106, −4.448455857621510, −3.962806727085152, −3.409278973581493, −2.956272872535936, −2.499789221821357, −1.857891463254851, −1.471203770557508, 0, 0,
1.471203770557508, 1.857891463254851, 2.499789221821357, 2.956272872535936, 3.409278973581493, 3.962806727085152, 4.448455857621510, 5.135444729366106, 5.428902477694703, 6.200457980775400, 6.471340044518876, 6.896279086350295, 7.429730091394514, 7.989807736965238, 8.480622617253920, 8.963524014733683, 9.470467024667608, 9.968039537823606, 10.27465325730215, 10.67675682443583, 11.32854403362590, 11.77466411807223, 12.45712709202838, 12.70727710446692, 12.99027345191888