| L(s) = 1 | + 2-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s − 11-s − 3·12-s − 2·13-s + 16-s − 7·17-s + 6·18-s + 5·19-s − 22-s + 6·23-s − 3·24-s − 2·26-s − 9·27-s − 4·31-s + 32-s + 3·33-s − 7·34-s + 6·36-s + 37-s + 5·38-s + 6·39-s − 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.301·11-s − 0.866·12-s − 0.554·13-s + 1/4·16-s − 1.69·17-s + 1.41·18-s + 1.14·19-s − 0.213·22-s + 1.25·23-s − 0.612·24-s − 0.392·26-s − 1.73·27-s − 0.718·31-s + 0.176·32-s + 0.522·33-s − 1.20·34-s + 36-s + 0.164·37-s + 0.811·38-s + 0.960·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−9.006497938916409149052258052203, −7.65898904963165500718515206931, −6.90390686229064882950913520290, −6.41920364978516898967942841242, −5.39344798256389985367504415125, −4.99317541960821719676613529042, −4.22473947000803998945761355947, −2.91348169870031232108756983593, −1.51366604578487137578114595906, 0,
1.51366604578487137578114595906, 2.91348169870031232108756983593, 4.22473947000803998945761355947, 4.99317541960821719676613529042, 5.39344798256389985367504415125, 6.41920364978516898967942841242, 6.90390686229064882950913520290, 7.65898904963165500718515206931, 9.006497938916409149052258052203
