L(s) = 1 | − 2·3-s − 5-s + 9-s + 2·15-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 4·27-s + 4·29-s + 31-s − 8·37-s − 6·41-s + 2·43-s − 45-s − 7·49-s + 4·51-s + 8·53-s + 8·57-s − 8·59-s − 4·67-s − 8·69-s − 6·73-s − 2·75-s − 4·79-s − 11·81-s + 6·83-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.516·15-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s + 0.742·29-s + 0.179·31-s − 1.31·37-s − 0.937·41-s + 0.304·43-s − 0.149·45-s − 49-s + 0.560·51-s + 1.09·53-s + 1.05·57-s − 1.04·59-s − 0.488·67-s − 0.963·69-s − 0.702·73-s − 0.230·75-s − 0.450·79-s − 1.22·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5038390472\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5038390472\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.36729834064092, −12.25090463040003, −11.84072468746887, −11.22676313082225, −10.89982361734627, −10.63513856559653, −10.06158976950811, −9.601955053419679, −8.791890962088835, −8.594044668308150, −8.188360525945400, −7.349500786838070, −6.985297487134551, −6.616559246922355, −6.065027199426754, −5.655757769415580, −5.032000559602360, −4.606328620685467, −4.315124734094336, −3.424579711377399, −3.076929226821011, −2.310634793625837, −1.642096270327605, −0.9411119709907348, −0.2436390643702702,
0.2436390643702702, 0.9411119709907348, 1.642096270327605, 2.310634793625837, 3.076929226821011, 3.424579711377399, 4.315124734094336, 4.606328620685467, 5.032000559602360, 5.655757769415580, 6.065027199426754, 6.616559246922355, 6.985297487134551, 7.349500786838070, 8.188360525945400, 8.594044668308150, 8.791890962088835, 9.601955053419679, 10.06158976950811, 10.63513856559653, 10.89982361734627, 11.22676313082225, 11.84072468746887, 12.25090463040003, 12.36729834064092