L(s) = 1 | + 2-s + 3-s + 4-s + 1.51·5-s + 6-s + 7-s + 8-s + 9-s + 1.51·10-s + 4.90·11-s + 12-s − 0.556·13-s + 14-s + 1.51·15-s + 16-s − 5.11·17-s + 18-s − 3.89·19-s + 1.51·20-s + 21-s + 4.90·22-s + 3.11·23-s + 24-s − 2.71·25-s − 0.556·26-s + 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.675·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.477·10-s + 1.47·11-s + 0.288·12-s − 0.154·13-s + 0.267·14-s + 0.390·15-s + 0.250·16-s − 1.24·17-s + 0.235·18-s − 0.894·19-s + 0.337·20-s + 0.218·21-s + 1.04·22-s + 0.648·23-s + 0.204·24-s − 0.543·25-s − 0.109·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.551500786\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.551500786\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 5 | \( 1 - 1.51T + 5T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 + 0.556T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 + 3.89T + 19T^{2} \) |
| 23 | \( 1 - 3.11T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 31 | \( 1 - 1.73T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 9.93T + 41T^{2} \) |
| 43 | \( 1 - 0.618T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 + 3.75T + 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 + 0.477T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 9.42T + 79T^{2} \) |
| 83 | \( 1 - 9.21T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.74960512849555698768320025868, −6.91855425052005401628332364692, −6.47439175364680231493325877445, −5.81778267793169761998641958526, −4.88042444812411292895959768176, −4.20669511733685076452075588870, −3.71604313368512360705304650340, −2.51262333418454686608523445471, −2.07759788637468851155298704579, −1.08163778619166860158613554722,
1.08163778619166860158613554722, 2.07759788637468851155298704579, 2.51262333418454686608523445471, 3.71604313368512360705304650340, 4.20669511733685076452075588870, 4.88042444812411292895959768176, 5.81778267793169761998641958526, 6.47439175364680231493325877445, 6.91855425052005401628332364692, 7.74960512849555698768320025868