L(s) = 1 | + 2-s + 1.91·3-s + 4-s − 2.55·5-s + 1.91·6-s + 0.891·7-s + 8-s + 0.682·9-s − 2.55·10-s − 2.68·11-s + 1.91·12-s − 0.516·13-s + 0.891·14-s − 4.89·15-s + 16-s − 1.21·17-s + 0.682·18-s + 3.53·19-s − 2.55·20-s + 1.71·21-s − 2.68·22-s + 23-s + 1.91·24-s + 1.50·25-s − 0.516·26-s − 4.44·27-s + 0.891·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.10·3-s + 0.5·4-s − 1.14·5-s + 0.783·6-s + 0.336·7-s + 0.353·8-s + 0.227·9-s − 0.806·10-s − 0.809·11-s + 0.554·12-s − 0.143·13-s + 0.238·14-s − 1.26·15-s + 0.250·16-s − 0.295·17-s + 0.160·18-s + 0.811·19-s − 0.570·20-s + 0.373·21-s − 0.572·22-s + 0.208·23-s + 0.391·24-s + 0.301·25-s − 0.101·26-s − 0.855·27-s + 0.168·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 5 | \( 1 + 2.55T + 5T^{2} \) |
| 7 | \( 1 - 0.891T + 7T^{2} \) |
| 11 | \( 1 + 2.68T + 11T^{2} \) |
| 13 | \( 1 + 0.516T + 13T^{2} \) |
| 17 | \( 1 + 1.21T + 17T^{2} \) |
| 19 | \( 1 - 3.53T + 19T^{2} \) |
| 29 | \( 1 + 0.777T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 41 | \( 1 + 8.13T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 - 4.49T + 59T^{2} \) |
| 61 | \( 1 + 4.71T + 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 3.20T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 6.23T + 83T^{2} \) |
| 89 | \( 1 - 3.68T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−7.66431286772745903551278470942, −7.31525803133942023564900212282, −6.37264890102527739948473003864, −5.20766941444008067266624729917, −4.88810742669657982069739977698, −3.63552192417240655540511827615, −3.52161712260642127556722883145, −2.57136009993835430205147235378, −1.70141901208101478297020941682, 0,
1.70141901208101478297020941682, 2.57136009993835430205147235378, 3.52161712260642127556722883145, 3.63552192417240655540511827615, 4.88810742669657982069739977698, 5.20766941444008067266624729917, 6.37264890102527739948473003864, 7.31525803133942023564900212282, 7.66431286772745903551278470942