L(s) = 1 | − 2.59·2-s − 3-s + 4.75·4-s − 3.50·5-s + 2.59·6-s + 0.252·7-s − 7.15·8-s + 9-s + 9.12·10-s + 11-s − 4.75·12-s + 0.314·13-s − 0.655·14-s + 3.50·15-s + 9.09·16-s + 3.13·17-s − 2.59·18-s + 0.841·19-s − 16.6·20-s − 0.252·21-s − 2.59·22-s + 1.45·23-s + 7.15·24-s + 7.31·25-s − 0.818·26-s − 27-s + 1.19·28-s + ⋯ |
L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.37·4-s − 1.56·5-s + 1.06·6-s + 0.0953·7-s − 2.53·8-s + 0.333·9-s + 2.88·10-s + 0.301·11-s − 1.37·12-s + 0.0873·13-s − 0.175·14-s + 0.906·15-s + 2.27·16-s + 0.759·17-s − 0.612·18-s + 0.192·19-s − 3.73·20-s − 0.0550·21-s − 0.554·22-s + 0.302·23-s + 1.46·24-s + 1.46·25-s − 0.160·26-s − 0.192·27-s + 0.226·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3082284533\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3082284533\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 5 | \( 1 + 3.50T + 5T^{2} \) |
| 7 | \( 1 - 0.252T + 7T^{2} \) |
| 13 | \( 1 - 0.314T + 13T^{2} \) |
| 17 | \( 1 - 3.13T + 17T^{2} \) |
| 19 | \( 1 - 0.841T + 19T^{2} \) |
| 23 | \( 1 - 1.45T + 23T^{2} \) |
| 29 | \( 1 + 6.76T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 8.89T + 37T^{2} \) |
| 41 | \( 1 + 8.43T + 41T^{2} \) |
| 43 | \( 1 - 1.80T + 43T^{2} \) |
| 47 | \( 1 - 2.30T + 47T^{2} \) |
| 53 | \( 1 + 4.41T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 67 | \( 1 + 0.0414T + 67T^{2} \) |
| 71 | \( 1 + 9.80T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 - 7.54T + 97T^{2} \) |
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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
−9.159021112684370405226971268858, −8.281939593239375157619580713115, −7.68847572228506357328687206936, −7.23415062957490701869006800735, −6.40544088829447325074783351353, −5.33613646063702112501229335142, −4.04395412211449067668976821401, −3.13112540859446497145040806868, −1.62557904602805132848864281664, −0.51206171655069171332020761223,
0.51206171655069171332020761223, 1.62557904602805132848864281664, 3.13112540859446497145040806868, 4.04395412211449067668976821401, 5.33613646063702112501229335142, 6.40544088829447325074783351353, 7.23415062957490701869006800735, 7.68847572228506357328687206936, 8.281939593239375157619580713115, 9.159021112684370405226971268858