L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s + 9-s + 2·10-s + 11-s − 12-s + 2·13-s + 2·15-s + 16-s − 17-s − 18-s − 4·19-s − 2·20-s − 22-s − 4·23-s + 24-s − 25-s − 2·26-s − 27-s − 6·29-s − 2·30-s + 8·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.554·13-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.365·30-s + 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6904789512\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6904789512\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.791639970413655917641576250492, −9.033013292171969391355411596080, −8.065145903852648498913401747313, −7.57088047211312208895685262066, −6.46235150203195023408990048323, −5.89843607773210695560034914230, −4.46660346048850343812953116696, −3.75518584215918539567619747473, −2.23819168515407055206657859405, −0.71150021091634101519329186610,
0.71150021091634101519329186610, 2.23819168515407055206657859405, 3.75518584215918539567619747473, 4.46660346048850343812953116696, 5.89843607773210695560034914230, 6.46235150203195023408990048323, 7.57088047211312208895685262066, 8.065145903852648498913401747313, 9.033013292171969391355411596080, 9.791639970413655917641576250492