L(s) = 1 | − 2.34·2-s − 3-s + 3.51·4-s + 1.78·5-s + 2.34·6-s + 7-s − 3.56·8-s + 9-s − 4.20·10-s + 4.79·11-s − 3.51·12-s − 1.06·13-s − 2.34·14-s − 1.78·15-s + 1.33·16-s − 6.56·17-s − 2.34·18-s + 1.65·19-s + 6.29·20-s − 21-s − 11.2·22-s − 4.89·23-s + 3.56·24-s − 1.79·25-s + 2.50·26-s − 27-s + 3.51·28-s + ⋯ |
L(s) = 1 | − 1.66·2-s − 0.577·3-s + 1.75·4-s + 0.800·5-s + 0.958·6-s + 0.377·7-s − 1.25·8-s + 0.333·9-s − 1.32·10-s + 1.44·11-s − 1.01·12-s − 0.295·13-s − 0.627·14-s − 0.462·15-s + 0.333·16-s − 1.59·17-s − 0.553·18-s + 0.380·19-s + 1.40·20-s − 0.218·21-s − 2.40·22-s − 1.02·23-s + 0.727·24-s − 0.359·25-s + 0.491·26-s − 0.192·27-s + 0.664·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 5 | \( 1 - 1.78T + 5T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 5.38T + 41T^{2} \) |
| 43 | \( 1 + 6.67T + 43T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + 8.82T + 53T^{2} \) |
| 59 | \( 1 + 1.10T + 59T^{2} \) |
| 61 | \( 1 + 9.20T + 61T^{2} \) |
| 67 | \( 1 + 4.37T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 8.79T + 73T^{2} \) |
| 79 | \( 1 + 1.77T + 79T^{2} \) |
| 83 | \( 1 + 5.39T + 83T^{2} \) |
| 89 | \( 1 + 7.92T + 89T^{2} \) |
| 97 | \( 1 + 6.39T + 97T^{2} \) |
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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
−7.64716104282857033299113795851, −6.73738742171677035935239510115, −6.42873987080197982337620204977, −5.75929701121831977468528098540, −4.66073929811130486951144455665, −4.00800224041793987460796924957, −2.54433530754281669659859196462, −1.83074942296325924188398453899, −1.18523122697561417811592324449, 0,
1.18523122697561417811592324449, 1.83074942296325924188398453899, 2.54433530754281669659859196462, 4.00800224041793987460796924957, 4.66073929811130486951144455665, 5.75929701121831977468528098540, 6.42873987080197982337620204977, 6.73738742171677035935239510115, 7.64716104282857033299113795851