L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 4·13-s + 2·14-s + 15-s + 16-s − 18-s − 4·19-s + 20-s − 2·21-s − 4·22-s − 8·23-s − 24-s + 25-s − 4·26-s + 27-s − 2·28-s − 2·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.436·21-s − 0.852·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.46056734779564941429858360470, −6.75073165539045686044172326517, −6.18187727537096742717053095158, −5.74172898428386676473715418926, −4.33022265459122172990971803694, −3.74654752082963807249745366918, −3.04645737126280039330654667295, −1.94628128059361467409829896058, −1.43936173499180391871452049440, 0,
1.43936173499180391871452049440, 1.94628128059361467409829896058, 3.04645737126280039330654667295, 3.74654752082963807249745366918, 4.33022265459122172990971803694, 5.74172898428386676473715418926, 6.18187727537096742717053095158, 6.75073165539045686044172326517, 7.46056734779564941429858360470