L(s) = 1 | + 2-s − 0.375·3-s + 4-s + 5-s − 0.375·6-s + 3.43·7-s + 8-s − 2.85·9-s + 10-s − 3.38·11-s − 0.375·12-s − 5.75·13-s + 3.43·14-s − 0.375·15-s + 16-s − 0.0635·17-s − 2.85·18-s − 1.00·19-s + 20-s − 1.28·21-s − 3.38·22-s − 3.87·23-s − 0.375·24-s + 25-s − 5.75·26-s + 2.19·27-s + 3.43·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.216·3-s + 0.5·4-s + 0.447·5-s − 0.153·6-s + 1.29·7-s + 0.353·8-s − 0.953·9-s + 0.316·10-s − 1.02·11-s − 0.108·12-s − 1.59·13-s + 0.918·14-s − 0.0968·15-s + 0.250·16-s − 0.0154·17-s − 0.673·18-s − 0.231·19-s + 0.223·20-s − 0.281·21-s − 0.721·22-s − 0.808·23-s − 0.0765·24-s + 0.200·25-s − 1.12·26-s + 0.423·27-s + 0.649·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.375T + 3T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 + 5.75T + 13T^{2} \) |
| 17 | \( 1 + 0.0635T + 17T^{2} \) |
| 19 | \( 1 + 1.00T + 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + 6.49T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 1.98T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 0.644T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 + 3.20T + 61T^{2} \) |
| 67 | \( 1 + 2.21T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 - 9.80T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 - 0.940T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 2.27T + 97T^{2} \) |
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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
−8.043855850233507402915108019423, −7.33569156674108602990265557868, −6.50617956494430731064101353299, −5.48015198379905790114079285935, −5.16878163874490625255096566892, −4.61540020948236489740475882547, −3.35879912011386665408072376316, −2.40499120838037847733085599115, −1.83063766644060458114333581584, 0,
1.83063766644060458114333581584, 2.40499120838037847733085599115, 3.35879912011386665408072376316, 4.61540020948236489740475882547, 5.16878163874490625255096566892, 5.48015198379905790114079285935, 6.50617956494430731064101353299, 7.33569156674108602990265557868, 8.043855850233507402915108019423