L(s) = 1 | − 2.27·2-s + 3-s + 3.15·4-s − 2.27·6-s + 0.797·7-s − 2.63·8-s + 9-s + 2.66·11-s + 3.15·12-s − 3.32·13-s − 1.81·14-s − 0.338·16-s − 2.55·17-s − 2.27·18-s + 4.89·19-s + 0.797·21-s − 6.06·22-s − 3.96·23-s − 2.63·24-s + 7.55·26-s + 27-s + 2.52·28-s − 7.42·29-s − 4.01·31-s + 6.03·32-s + 2.66·33-s + 5.80·34-s + ⋯ |
L(s) = 1 | − 1.60·2-s + 0.577·3-s + 1.57·4-s − 0.927·6-s + 0.301·7-s − 0.930·8-s + 0.333·9-s + 0.804·11-s + 0.911·12-s − 0.922·13-s − 0.484·14-s − 0.0847·16-s − 0.620·17-s − 0.535·18-s + 1.12·19-s + 0.174·21-s − 1.29·22-s − 0.827·23-s − 0.537·24-s + 1.48·26-s + 0.192·27-s + 0.476·28-s − 1.37·29-s − 0.721·31-s + 1.06·32-s + 0.464·33-s + 0.996·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 7 | \( 1 - 0.797T + 7T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 + 3.32T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 + 7.42T + 29T^{2} \) |
| 31 | \( 1 + 4.01T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 + 8.81T + 41T^{2} \) |
| 43 | \( 1 + 6.51T + 43T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 - 0.913T + 61T^{2} \) |
| 67 | \( 1 + 5.01T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7.63T + 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−8.122065678628273840832408099124, −7.80094391083951985522768829002, −7.04255279608272917265515866732, −6.39125169798531997319882171739, −5.19507726226053125909902174399, −4.23350117505392028683980169342, −3.17941280269483484402300124961, −2.09213612539067359217404046582, −1.44060144039195679427993956419, 0,
1.44060144039195679427993956419, 2.09213612539067359217404046582, 3.17941280269483484402300124961, 4.23350117505392028683980169342, 5.19507726226053125909902174399, 6.39125169798531997319882171739, 7.04255279608272917265515866732, 7.80094391083951985522768829002, 8.122065678628273840832408099124