| L(s) = 1 | + 1.30·3-s − 3·5-s + 0.302·7-s − 1.30·9-s + 2.69·13-s − 3.90·15-s + 2.30·17-s − 2·19-s + 0.394·21-s + 2.30·23-s + 4·25-s − 5.60·27-s + 7.60·29-s − 6.60·31-s − 0.908·35-s + 0.394·37-s + 3.51·39-s − 6.21·41-s − 9.60·43-s + 3.90·45-s − 1.60·47-s − 6.90·49-s + 3·51-s − 4.60·53-s − 2.60·57-s − 7.81·59-s + 6.81·61-s + ⋯ |
| L(s) = 1 | + 0.752·3-s − 1.34·5-s + 0.114·7-s − 0.434·9-s + 0.748·13-s − 1.00·15-s + 0.558·17-s − 0.458·19-s + 0.0860·21-s + 0.480·23-s + 0.800·25-s − 1.07·27-s + 1.41·29-s − 1.18·31-s − 0.153·35-s + 0.0648·37-s + 0.562·39-s − 0.970·41-s − 1.46·43-s + 0.582·45-s − 0.234·47-s − 0.986·49-s + 0.420·51-s − 0.632·53-s − 0.345·57-s − 1.01·59-s + 0.872·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 - 0.302T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 2.30T + 23T^{2} \) |
| 29 | \( 1 - 7.60T + 29T^{2} \) |
| 31 | \( 1 + 6.60T + 31T^{2} \) |
| 37 | \( 1 - 0.394T + 37T^{2} \) |
| 41 | \( 1 + 6.21T + 41T^{2} \) |
| 43 | \( 1 + 9.60T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 4.60T + 53T^{2} \) |
| 59 | \( 1 + 7.81T + 59T^{2} \) |
| 61 | \( 1 - 6.81T + 61T^{2} \) |
| 67 | \( 1 + 8.21T + 67T^{2} \) |
| 71 | \( 1 + 3.90T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 4.51T + 97T^{2} \) |
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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
−8.382725397743591804389177484016, −7.958114423324402655419227903907, −7.13568508343499727524962763224, −6.28536557366232249434706804624, −5.22514132203385272999856377627, −4.31283093638949506405694297817, −3.45989368881514867852212270543, −2.98178009582694149725519491649, −1.55807522310230759260450286225, 0,
1.55807522310230759260450286225, 2.98178009582694149725519491649, 3.45989368881514867852212270543, 4.31283093638949506405694297817, 5.22514132203385272999856377627, 6.28536557366232249434706804624, 7.13568508343499727524962763224, 7.958114423324402655419227903907, 8.382725397743591804389177484016
