L(s) = 1 | − 1.40·2-s + 3-s − 0.0314·4-s + 1.13·5-s − 1.40·6-s − 1.90·7-s + 2.85·8-s + 9-s − 1.58·10-s + 5.04·11-s − 0.0314·12-s + 3.78·13-s + 2.67·14-s + 1.13·15-s − 3.93·16-s + 7.36·17-s − 1.40·18-s − 0.713·19-s − 0.0356·20-s − 1.90·21-s − 7.07·22-s − 23-s + 2.85·24-s − 3.71·25-s − 5.30·26-s + 27-s + 0.0600·28-s + ⋯ |
L(s) = 1 | − 0.992·2-s + 0.577·3-s − 0.0157·4-s + 0.506·5-s − 0.572·6-s − 0.721·7-s + 1.00·8-s + 0.333·9-s − 0.502·10-s + 1.52·11-s − 0.00907·12-s + 1.04·13-s + 0.716·14-s + 0.292·15-s − 0.984·16-s + 1.78·17-s − 0.330·18-s − 0.163·19-s − 0.00796·20-s − 0.416·21-s − 1.50·22-s − 0.208·23-s + 0.581·24-s − 0.743·25-s − 1.04·26-s + 0.192·27-s + 0.0113·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.459741105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.459741105\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 + 1.40T + 2T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 - 7.36T + 17T^{2} \) |
| 19 | \( 1 + 0.713T + 19T^{2} \) |
| 31 | \( 1 - 4.34T + 31T^{2} \) |
| 37 | \( 1 - 7.55T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 1.16T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 - 4.46T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 0.947T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 9.66T + 89T^{2} \) |
| 97 | \( 1 + 3.19T + 97T^{2} \) |
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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
−9.408704508830593823985016348084, −8.271869127739027741606331317693, −8.109745547409813068009115395385, −6.79817197583708999485858161304, −6.32722757187325960684571620142, −5.19411492042208535284766007590, −3.90279774286387843458043547293, −3.39178156488039466792972637136, −1.82372889318151553061879159908, −1.00298500817664978434868462515,
1.00298500817664978434868462515, 1.82372889318151553061879159908, 3.39178156488039466792972637136, 3.90279774286387843458043547293, 5.19411492042208535284766007590, 6.32722757187325960684571620142, 6.79817197583708999485858161304, 8.109745547409813068009115395385, 8.271869127739027741606331317693, 9.408704508830593823985016348084