L(s) = 1 | − 1.76·2-s − 3-s + 1.12·4-s + 2.69·5-s + 1.76·6-s − 0.714·7-s + 1.54·8-s + 9-s − 4.76·10-s − 1.68·11-s − 1.12·12-s + 2.54·13-s + 1.26·14-s − 2.69·15-s − 4.98·16-s + 17-s − 1.76·18-s + 6.37·19-s + 3.03·20-s + 0.714·21-s + 2.96·22-s + 3.24·23-s − 1.54·24-s + 2.26·25-s − 4.49·26-s − 27-s − 0.803·28-s + ⋯ |
L(s) = 1 | − 1.25·2-s − 0.577·3-s + 0.562·4-s + 1.20·5-s + 0.721·6-s − 0.269·7-s + 0.546·8-s + 0.333·9-s − 1.50·10-s − 0.506·11-s − 0.324·12-s + 0.704·13-s + 0.337·14-s − 0.695·15-s − 1.24·16-s + 0.242·17-s − 0.416·18-s + 1.46·19-s + 0.678·20-s + 0.155·21-s + 0.633·22-s + 0.676·23-s − 0.315·24-s + 0.453·25-s − 0.881·26-s − 0.192·27-s − 0.151·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.011860542\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011860542\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 1.76T + 2T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 7 | \( 1 + 0.714T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 19 | \( 1 - 6.37T + 19T^{2} \) |
| 23 | \( 1 - 3.24T + 23T^{2} \) |
| 29 | \( 1 - 3.15T + 29T^{2} \) |
| 31 | \( 1 - 2.83T + 31T^{2} \) |
| 37 | \( 1 + 8.46T + 37T^{2} \) |
| 41 | \( 1 + 8.96T + 41T^{2} \) |
| 43 | \( 1 - 6.33T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 - 9.52T + 53T^{2} \) |
| 59 | \( 1 - 5.83T + 59T^{2} \) |
| 61 | \( 1 - 5.40T + 61T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 - 1.99T + 89T^{2} \) |
| 97 | \( 1 + 3.32T + 97T^{2} \) |
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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
−8.587335424225014256872941601478, −7.83629947669138853873706927908, −6.97059664514536164876689250219, −6.45904501805548324305157874628, −5.39154089309077952508751431155, −5.12559568753328524931036825006, −3.75461215572213947469587940928, −2.60674211772110275350608663947, −1.55928063622633860842676021251, −0.77633042787671057989212222239,
0.77633042787671057989212222239, 1.55928063622633860842676021251, 2.60674211772110275350608663947, 3.75461215572213947469587940928, 5.12559568753328524931036825006, 5.39154089309077952508751431155, 6.45904501805548324305157874628, 6.97059664514536164876689250219, 7.83629947669138853873706927908, 8.587335424225014256872941601478