| L(s) = 1 | − 1.16·3-s − 4.97·7-s − 1.63·9-s + 11-s − 0.665·13-s + 6.77·17-s − 19-s + 5.80·21-s − 2.16·23-s + 5.41·27-s + 7.97·29-s + 8.94·31-s − 1.16·33-s − 0.139·37-s + 0.776·39-s − 1.80·41-s − 2.80·43-s − 0.530·47-s + 17.7·49-s − 7.91·51-s − 6.30·53-s + 1.16·57-s − 11.4·59-s + 10.5·61-s + 8.13·63-s − 9.60·67-s + 2.53·69-s + ⋯ |
| L(s) = 1 | − 0.674·3-s − 1.87·7-s − 0.545·9-s + 0.301·11-s − 0.184·13-s + 1.64·17-s − 0.229·19-s + 1.26·21-s − 0.451·23-s + 1.04·27-s + 1.48·29-s + 1.60·31-s − 0.203·33-s − 0.0229·37-s + 0.124·39-s − 0.281·41-s − 0.427·43-s − 0.0773·47-s + 2.53·49-s − 1.10·51-s − 0.865·53-s + 0.154·57-s − 1.49·59-s + 1.35·61-s + 1.02·63-s − 1.17·67-s + 0.304·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.16T + 3T^{2} \) |
| 7 | \( 1 + 4.97T + 7T^{2} \) |
| 13 | \( 1 + 0.665T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 + 0.139T + 37T^{2} \) |
| 41 | \( 1 + 1.80T + 41T^{2} \) |
| 43 | \( 1 + 2.80T + 43T^{2} \) |
| 47 | \( 1 + 0.530T + 47T^{2} \) |
| 53 | \( 1 + 6.30T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 9.60T + 67T^{2} \) |
| 71 | \( 1 - 9.80T + 71T^{2} \) |
| 73 | \( 1 + 7.02T + 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 9.58T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.043540024727641804896712990976, −7.02592286658170180911874833097, −6.35986554270084087487157755702, −6.01172807671707852095711789840, −5.18825381742332467794144550096, −4.19981592824679162461799832749, −3.16319403514442045040955733300, −2.80334525384440957543101206241, −1.07289021345290563485646633503, 0,
1.07289021345290563485646633503, 2.80334525384440957543101206241, 3.16319403514442045040955733300, 4.19981592824679162461799832749, 5.18825381742332467794144550096, 6.01172807671707852095711789840, 6.35986554270084087487157755702, 7.02592286658170180911874833097, 8.043540024727641804896712990976
