L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 6·11-s − 12-s − 4·13-s − 14-s + 15-s + 16-s − 18-s − 20-s − 21-s + 6·22-s − 4·23-s + 24-s + 25-s + 4·26-s − 27-s + 28-s + 8·29-s − 30-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 − 1.80·11-s − 0.288·12-s − 1.10·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 1.27·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s + 0.188·28-s + 1.48·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5487195046\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5487195046\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.37861853672323, −13.70736351474609, −13.31158270706818, −12.41810645728389, −12.29174546206012, −11.79082858115320, −11.15531769498471, −10.60057077315859, −10.30632215030625, −9.863609662422231, −9.231322930916491, −8.362300593963862, −8.154744434511382, −7.468178120884475, −7.299383759645884, −6.438843714955260, −5.865073216783286, −5.285985337761413, −4.656697388936133, −4.330771834637054, −3.199401239018988, −2.591830059751172, −2.152448753253806, −1.072715816166305, −0.3316464125145600,
0.3316464125145600, 1.072715816166305, 2.152448753253806, 2.591830059751172, 3.199401239018988, 4.330771834637054, 4.656697388936133, 5.285985337761413, 5.865073216783286, 6.438843714955260, 7.299383759645884, 7.468178120884475, 8.154744434511382, 8.362300593963862, 9.231322930916491, 9.863609662422231, 10.30632215030625, 10.60057077315859, 11.15531769498471, 11.79082858115320, 12.29174546206012, 12.41810645728389, 13.31158270706818, 13.70736351474609, 14.37861853672323