L(s) = 1 | + 2-s + 4-s − 2·5-s + 4·7-s + 8-s − 2·10-s − 11-s + 6·13-s + 4·14-s + 16-s + 17-s + 4·19-s − 2·20-s − 22-s − 25-s + 6·26-s + 4·28-s − 2·29-s + 32-s + 34-s − 8·35-s − 10·37-s + 4·38-s − 2·40-s + 10·41-s − 4·43-s − 44-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s + 0.353·8-s − 0.632·10-s − 0.301·11-s + 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.447·20-s − 0.213·22-s − 1/5·25-s + 1.17·26-s + 0.755·28-s − 0.371·29-s + 0.176·32-s + 0.171·34-s − 1.35·35-s − 1.64·37-s + 0.648·38-s − 0.316·40-s + 1.56·41-s − 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.344196509\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.344196509\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.349008443648270540657545078133, −7.85771500700694880481704031415, −7.27626020256065065424350192262, −6.23140998788600193607778058300, −5.40395867781671938243990655247, −4.80862388855917089768646831397, −3.87717055696743372397518874491, −3.39626032699524658965767129129, −2.02628595811140706792910814479, −1.06452464631268195421122058811,
1.06452464631268195421122058811, 2.02628595811140706792910814479, 3.39626032699524658965767129129, 3.87717055696743372397518874491, 4.80862388855917089768646831397, 5.40395867781671938243990655247, 6.23140998788600193607778058300, 7.27626020256065065424350192262, 7.85771500700694880481704031415, 8.349008443648270540657545078133