| L(s) = 1 | + 3-s − 5-s − 4.77·7-s + 9-s + 3.30·11-s + 3.19·13-s − 15-s − 2.41·17-s + 2.24·19-s − 4.77·21-s + 23-s + 25-s + 27-s − 7.96·29-s − 10.3·31-s + 3.30·33-s + 4.77·35-s − 11.2·37-s + 3.19·39-s + 1.02·41-s + 10.0·43-s − 45-s + 2.74·47-s + 15.8·49-s − 2.41·51-s − 5.58·53-s − 3.30·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.80·7-s + 0.333·9-s + 0.997·11-s + 0.884·13-s − 0.258·15-s − 0.585·17-s + 0.515·19-s − 1.04·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.47·29-s − 1.85·31-s + 0.575·33-s + 0.807·35-s − 1.85·37-s + 0.510·39-s + 0.160·41-s + 1.53·43-s − 0.149·45-s + 0.400·47-s + 2.26·49-s − 0.337·51-s − 0.767·53-s − 0.446·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 4.77T + 7T^{2} \) |
| 11 | \( 1 - 3.30T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 29 | \( 1 + 7.96T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 1.02T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 + 5.58T + 53T^{2} \) |
| 59 | \( 1 + 1.96T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 7.71T + 67T^{2} \) |
| 71 | \( 1 - 16.0T + 71T^{2} \) |
| 73 | \( 1 + 7.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 3.05T + 89T^{2} \) |
| 97 | \( 1 + 5.43T + 97T^{2} \) |
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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
−8.740334807442024627108787971146, −7.50510474876520823398869112960, −6.99918363048691458253743302802, −6.26805090099508634182308157447, −5.48738686468975355876507107614, −3.96107618347204100650787091529, −3.71270115690517502250482912992, −2.87708540364461310259897329174, −1.54602499741009197119441673069, 0,
1.54602499741009197119441673069, 2.87708540364461310259897329174, 3.71270115690517502250482912992, 3.96107618347204100650787091529, 5.48738686468975355876507107614, 6.26805090099508634182308157447, 6.99918363048691458253743302802, 7.50510474876520823398869112960, 8.740334807442024627108787971146
