L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 5-s + 2·6-s + 7-s + 9-s − 2·10-s − 5·11-s + 2·12-s + 2·14-s − 15-s − 4·16-s + 6·17-s + 2·18-s + 19-s − 2·20-s + 21-s − 10·22-s + 25-s + 27-s + 2·28-s − 2·30-s + 2·31-s − 8·32-s − 5·33-s + 12·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s − 1.50·11-s + 0.577·12-s + 0.534·14-s − 0.258·15-s − 16-s + 1.45·17-s + 0.471·18-s + 0.229·19-s − 0.447·20-s + 0.218·21-s − 2.13·22-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.365·30-s + 0.359·31-s − 1.41·32-s − 0.870·33-s + 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55545 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.007156352\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.007156352\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.32433111377181, −13.93105711563026, −13.34027057197386, −13.05516897389590, −12.48659447696281, −11.99350735752918, −11.61000532537530, −10.92169742888171, −10.41267167665765, −9.805284238219745, −9.319006200903385, −8.374136542958104, −8.065421643320768, −7.684722001867782, −6.924403549864347, −6.397840343676247, −5.604829568781497, −5.045904213651694, −4.913158278021140, −4.038956914524903, −3.382682731789818, −3.109617806389261, −2.407325104055699, −1.681744548934568, −0.5643382253541548,
0.5643382253541548, 1.681744548934568, 2.407325104055699, 3.109617806389261, 3.382682731789818, 4.038956914524903, 4.913158278021140, 5.045904213651694, 5.604829568781497, 6.397840343676247, 6.924403549864347, 7.684722001867782, 8.065421643320768, 8.374136542958104, 9.319006200903385, 9.805284238219745, 10.41267167665765, 10.92169742888171, 11.61000532537530, 11.99350735752918, 12.48659447696281, 13.05516897389590, 13.34027057197386, 13.93105711563026, 14.32433111377181