| L(s) = 1 | + 2·13-s + 3·17-s + 8·19-s + 9·23-s + 6·29-s + 5·31-s − 8·37-s − 3·41-s − 10·43-s − 3·47-s + 6·53-s − 12·59-s + 4·61-s + 2·67-s − 9·71-s − 10·73-s − 5·79-s − 6·83-s + 3·89-s + 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.554·13-s + 0.727·17-s + 1.83·19-s + 1.87·23-s + 1.11·29-s + 0.898·31-s − 1.31·37-s − 0.468·41-s − 1.52·43-s − 0.437·47-s + 0.824·53-s − 1.56·59-s + 0.512·61-s + 0.244·67-s − 1.06·71-s − 1.17·73-s − 0.562·79-s − 0.658·83-s + 0.317·89-s + 0.507·97-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
| \(L(1)\) |
\(\approx\) |
\(3.314167844\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.314167844\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−13.28199433484527, −12.74616567697272, −12.03633135191403, −11.86574952929559, −11.41176102308628, −10.78540088263948, −10.26119782173144, −9.976702957708169, −9.358296215444741, −8.854849894517837, −8.449121562324304, −7.903036184601249, −7.366982536522519, −6.805506269734951, −6.573830590637992, −5.636294509441465, −5.393312229045960, −4.826950880581678, −4.331035310309857, −3.362616215059501, −3.192039224322309, −2.733138135947200, −1.568160034694044, −1.276474488414172, −0.5661025341812445,
0.5661025341812445, 1.276474488414172, 1.568160034694044, 2.733138135947200, 3.192039224322309, 3.362616215059501, 4.331035310309857, 4.826950880581678, 5.393312229045960, 5.636294509441465, 6.573830590637992, 6.805506269734951, 7.366982536522519, 7.903036184601249, 8.449121562324304, 8.854849894517837, 9.358296215444741, 9.976702957708169, 10.26119782173144, 10.78540088263948, 11.41176102308628, 11.86574952929559, 12.03633135191403, 12.74616567697272, 13.28199433484527
