| L(s) = 1 | + 1.41·3-s + 2.51·5-s − 0.732·7-s − 0.999·9-s + 0.732·11-s − 4.86·13-s + 3.56·15-s − 6.44·17-s − 5.56·19-s − 1.03·21-s + 1.33·25-s − 5.65·27-s + 6.42·29-s + 2.49·31-s + 1.03·33-s − 1.84·35-s + 2.74·37-s − 6.87·39-s + 2.46·41-s + 2.02·43-s − 2.51·45-s − 8.71·47-s − 6.46·49-s − 9.12·51-s + 3.08·53-s + 1.84·55-s − 7.86·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 1.12·5-s − 0.276·7-s − 0.333·9-s + 0.220·11-s − 1.34·13-s + 0.919·15-s − 1.56·17-s − 1.27·19-s − 0.225·21-s + 0.267·25-s − 1.08·27-s + 1.19·29-s + 0.448·31-s + 0.180·33-s − 0.311·35-s + 0.450·37-s − 1.10·39-s + 0.384·41-s + 0.309·43-s − 0.375·45-s − 1.27·47-s − 0.923·49-s − 1.27·51-s + 0.424·53-s + 0.248·55-s − 1.04·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.51T + 5T^{2} \) |
| 7 | \( 1 + 0.732T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 13 | \( 1 + 4.86T + 13T^{2} \) |
| 17 | \( 1 + 6.44T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 29 | \( 1 - 6.42T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 - 2.74T + 37T^{2} \) |
| 41 | \( 1 - 2.46T + 41T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.56T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7.26T + 73T^{2} \) |
| 79 | \( 1 - 17.1T + 79T^{2} \) |
| 83 | \( 1 - 5.50T + 83T^{2} \) |
| 89 | \( 1 + 5.69T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−8.166757094488695914158619051368, −7.34313091128168652078972267743, −6.34886983356496919097272887980, −6.14243855081476493347962219285, −4.87758378995380154587423811285, −4.36510380498480280803379381843, −3.06059716697715972538096790002, −2.43751291473609325197038726479, −1.83036306716282595731336402118, 0,
1.83036306716282595731336402118, 2.43751291473609325197038726479, 3.06059716697715972538096790002, 4.36510380498480280803379381843, 4.87758378995380154587423811285, 6.14243855081476493347962219285, 6.34886983356496919097272887980, 7.34313091128168652078972267743, 8.166757094488695914158619051368
