L(s) = 1 | − 2.65·2-s − 3-s + 5.03·4-s + 3.62·5-s + 2.65·6-s − 1.69·7-s − 8.06·8-s + 9-s − 9.62·10-s + 0.122·11-s − 5.03·12-s + 13-s + 4.50·14-s − 3.62·15-s + 11.3·16-s + 1.64·17-s − 2.65·18-s + 2.48·19-s + 18.2·20-s + 1.69·21-s − 0.325·22-s + 7.71·23-s + 8.06·24-s + 8.14·25-s − 2.65·26-s − 27-s − 8.55·28-s + ⋯ |
L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.51·4-s + 1.62·5-s + 1.08·6-s − 0.641·7-s − 2.85·8-s + 0.333·9-s − 3.04·10-s + 0.0369·11-s − 1.45·12-s + 0.277·13-s + 1.20·14-s − 0.936·15-s + 2.82·16-s + 0.399·17-s − 0.625·18-s + 0.570·19-s + 4.08·20-s + 0.370·21-s − 0.0693·22-s + 1.60·23-s + 1.64·24-s + 1.62·25-s − 0.520·26-s − 0.192·27-s − 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8833075407\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8833075407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 - 3.62T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 - 0.122T + 11T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 - 7.71T + 23T^{2} \) |
| 29 | \( 1 + 8.37T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 5.29T + 41T^{2} \) |
| 43 | \( 1 - 5.82T + 43T^{2} \) |
| 47 | \( 1 - 11.9T + 47T^{2} \) |
| 53 | \( 1 + 3.27T + 53T^{2} \) |
| 59 | \( 1 - 9.14T + 59T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 6.39T + 73T^{2} \) |
| 79 | \( 1 - 9.16T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 - 1.34T + 89T^{2} \) |
| 97 | \( 1 - 8.30T + 97T^{2} \) |
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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
−8.838945727981487872032771073719, −7.62050216992773679361166762160, −7.15582339612203780987701925291, −6.30579036571721507470204211305, −5.89986463181133977521305231434, −5.14564733171739913917542697349, −3.39786250515803182705520575721, −2.46703708838501200953500525247, −1.60121592086931748901383574825, −0.76985518137393737681112426385,
0.76985518137393737681112426385, 1.60121592086931748901383574825, 2.46703708838501200953500525247, 3.39786250515803182705520575721, 5.14564733171739913917542697349, 5.89986463181133977521305231434, 6.30579036571721507470204211305, 7.15582339612203780987701925291, 7.62050216992773679361166762160, 8.838945727981487872032771073719