L(s) = 1 | − 2-s − 3.13·3-s + 4-s + 5-s + 3.13·6-s + 2.49·7-s − 8-s + 6.79·9-s − 10-s − 1.60·11-s − 3.13·12-s − 0.554·13-s − 2.49·14-s − 3.13·15-s + 16-s + 5.26·17-s − 6.79·18-s − 2.32·19-s + 20-s − 7.82·21-s + 1.60·22-s + 4.26·23-s + 3.13·24-s + 25-s + 0.554·26-s − 11.8·27-s + 2.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.80·3-s + 0.5·4-s + 0.447·5-s + 1.27·6-s + 0.944·7-s − 0.353·8-s + 2.26·9-s − 0.316·10-s − 0.483·11-s − 0.903·12-s − 0.153·13-s − 0.667·14-s − 0.808·15-s + 0.250·16-s + 1.27·17-s − 1.60·18-s − 0.533·19-s + 0.223·20-s − 1.70·21-s + 0.341·22-s + 0.889·23-s + 0.638·24-s + 0.200·25-s + 0.108·26-s − 2.28·27-s + 0.472·28-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 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 1.60T + 11T^{2} \) |
| 13 | \( 1 + 0.554T + 13T^{2} \) |
| 17 | \( 1 - 5.26T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 - 4.26T + 23T^{2} \) |
| 29 | \( 1 + 2.40T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 0.359T + 37T^{2} \) |
| 41 | \( 1 + 2.50T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 6.02T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 - 0.692T + 61T^{2} \) |
| 67 | \( 1 + 3.85T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 - 5.75T + 73T^{2} \) |
| 79 | \( 1 - 1.29T + 79T^{2} \) |
| 83 | \( 1 - 2.60T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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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
−7.81596506617252227213903634703, −7.42215725632832677856535560128, −6.53323599808328707839647432965, −5.85353182001760581489939407335, −5.18921943059932567154571545393, −4.74815286744035664864617721149, −3.42097093133713388531807582139, −1.91963133058607659587934320137, −1.22402491260016228744459577363, 0,
1.22402491260016228744459577363, 1.91963133058607659587934320137, 3.42097093133713388531807582139, 4.74815286744035664864617721149, 5.18921943059932567154571545393, 5.85353182001760581489939407335, 6.53323599808328707839647432965, 7.42215725632832677856535560128, 7.81596506617252227213903634703