L(s) = 1 | − 2-s + 3-s + 4-s + 3.07·5-s − 6-s − 1.20·7-s − 8-s + 9-s − 3.07·10-s + 1.64·11-s + 12-s − 13-s + 1.20·14-s + 3.07·15-s + 16-s + 5.16·17-s − 18-s − 6.97·19-s + 3.07·20-s − 1.20·21-s − 1.64·22-s − 4.45·23-s − 24-s + 4.42·25-s + 26-s + 27-s − 1.20·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.37·5-s − 0.408·6-s − 0.454·7-s − 0.353·8-s + 0.333·9-s − 0.970·10-s + 0.495·11-s + 0.288·12-s − 0.277·13-s + 0.321·14-s + 0.792·15-s + 0.250·16-s + 1.25·17-s − 0.235·18-s − 1.59·19-s + 0.686·20-s − 0.262·21-s − 0.350·22-s − 0.928·23-s − 0.204·24-s + 0.885·25-s + 0.196·26-s + 0.192·27-s − 0.227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 + 4.45T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 + 0.105T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 1.62T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 + 9.64T + 47T^{2} \) |
| 53 | \( 1 + 8.08T + 53T^{2} \) |
| 59 | \( 1 + 4.90T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 0.0855T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 - 4.12T + 89T^{2} \) |
| 97 | \( 1 + 3.96T + 97T^{2} \) |
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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
−7.64519394056587568895200002042, −6.66054606803140029720774184544, −6.33674976426227144088012389643, −5.61240652750012187399024408398, −4.72837202351113348525910983734, −3.61019049402305205129348588271, −2.98541251421955017385538127518, −1.79041168928335264322328387509, −1.73449971740347984951326747604, 0,
1.73449971740347984951326747604, 1.79041168928335264322328387509, 2.98541251421955017385538127518, 3.61019049402305205129348588271, 4.72837202351113348525910983734, 5.61240652750012187399024408398, 6.33674976426227144088012389643, 6.66054606803140029720774184544, 7.64519394056587568895200002042