L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 12-s − 13-s + 14-s + 15-s + 16-s − 6·17-s − 18-s + 4·19-s + 20-s − 21-s − 24-s + 25-s + 26-s + 27-s − 28-s − 6·29-s − 30-s − 4·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.20754010116980, −12.56961257920676, −12.08492558065760, −11.74113876371290, −11.00524085902890, −10.75293389615186, −10.27412389579026, −9.698245091754057, −9.371018775245348, −8.980215554036355, −8.667297549498644, −8.014914381220274, −7.522505934050622, −7.134124378377973, −6.642286927297682, −6.248479306589172, −5.596933833325733, −5.047789650601183, −4.597241918532166, −3.822744249397698, −3.249638684757378, −2.973918551822306, −2.123402041635633, −1.781417864120688, −1.296925971340996, 0, 0,
1.296925971340996, 1.781417864120688, 2.123402041635633, 2.973918551822306, 3.249638684757378, 3.822744249397698, 4.597241918532166, 5.047789650601183, 5.596933833325733, 6.248479306589172, 6.642286927297682, 7.134124378377973, 7.522505934050622, 8.014914381220274, 8.667297549498644, 8.980215554036355, 9.371018775245348, 9.698245091754057, 10.27412389579026, 10.75293389615186, 11.00524085902890, 11.74113876371290, 12.08492558065760, 12.56961257920676, 13.20754010116980