L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 8-s + 9-s − 4·10-s − 11-s + 12-s − 13-s − 4·15-s + 16-s + 3·17-s + 18-s − 19-s − 4·20-s − 22-s + 6·23-s + 24-s + 11·25-s − 26-s + 27-s − 9·29-s − 4·30-s − 8·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s − 1.03·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.894·20-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.196·26-s + 0.192·27-s − 1.67·29-s − 0.730·30-s − 1.43·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.83546832108687983149480687471, −7.45564495470119887695088578982, −6.96128632507494091905434744345, −5.71791062407358755536757733730, −4.89934602440930768038535189510, −4.17162893266790488479124875627, −3.43558600395004202156010299453, −2.96606534953485372636882370179, −1.58923891034257184813946926122, 0,
1.58923891034257184813946926122, 2.96606534953485372636882370179, 3.43558600395004202156010299453, 4.17162893266790488479124875627, 4.89934602440930768038535189510, 5.71791062407358755536757733730, 6.96128632507494091905434744345, 7.45564495470119887695088578982, 7.83546832108687983149480687471