L(s) = 1 | − 2·11-s + 4·13-s − 5·17-s + 4·19-s − 5·23-s + 6·29-s + 11·31-s − 8·37-s − 5·41-s + 47-s + 12·53-s − 2·59-s + 10·61-s − 71-s − 2·73-s − 9·79-s − 6·83-s − 11·89-s + 97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.603·11-s + 1.10·13-s − 1.21·17-s + 0.917·19-s − 1.04·23-s + 1.11·29-s + 1.97·31-s − 1.31·37-s − 0.780·41-s + 0.145·47-s + 1.64·53-s − 0.260·59-s + 1.28·61-s − 0.118·71-s − 0.234·73-s − 1.01·79-s − 0.658·83-s − 1.16·89-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.238072415\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.238072415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.37292628347877, −12.78817586076016, −12.03291348143629, −11.86449742993905, −11.37300919740974, −10.74758048843346, −10.30635904522305, −10.00776026216554, −9.424129342020644, −8.640242445442533, −8.404033569039296, −8.175235741925419, −7.206976594343447, −6.968614191915620, −6.331937035100399, −5.890438876079253, −5.337049559384925, −4.743886173570852, −4.238142989227322, −3.707158615614473, −3.013573343615233, −2.551802796815296, −1.855561494262083, −1.155612352371357, −0.4594255347924531,
0.4594255347924531, 1.155612352371357, 1.855561494262083, 2.551802796815296, 3.013573343615233, 3.707158615614473, 4.238142989227322, 4.743886173570852, 5.337049559384925, 5.890438876079253, 6.331937035100399, 6.968614191915620, 7.206976594343447, 8.175235741925419, 8.404033569039296, 8.640242445442533, 9.424129342020644, 10.00776026216554, 10.30635904522305, 10.74758048843346, 11.37300919740974, 11.86449742993905, 12.03291348143629, 12.78817586076016, 13.37292628347877