| L(s) = 1 | − 2·3-s − 4·7-s + 9-s + 11-s + 4·13-s − 4·19-s + 8·21-s − 6·23-s + 4·27-s + 6·29-s − 8·31-s − 2·33-s + 2·37-s − 8·39-s − 6·41-s − 8·43-s − 6·47-s + 9·49-s + 6·53-s + 8·57-s − 12·59-s − 2·61-s − 4·63-s + 10·67-s + 12·69-s + 12·71-s − 16·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s − 0.917·19-s + 1.74·21-s − 1.25·23-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.348·33-s + 0.328·37-s − 1.28·39-s − 0.937·41-s − 1.21·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.05·57-s − 1.56·59-s − 0.256·61-s − 0.503·63-s + 1.22·67-s + 1.44·69-s + 1.42·71-s − 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.80206643818519, −12.39710737743799, −11.89365269429954, −11.60178527287197, −11.00843535570219, −10.57654373846570, −10.26600596657329, −9.739957229709472, −9.334482213088615, −8.609485278189752, −8.448984103220546, −7.774920496346804, −6.892110854579573, −6.747504393408174, −6.196074168211381, −6.031565412744854, −5.473504133404292, −4.886737073814798, −4.244936269349019, −3.765426944214685, −3.302418177993942, −2.751700500332949, −1.911410435621373, −1.347362911336997, −0.4825516685335618, 0,
0.4825516685335618, 1.347362911336997, 1.911410435621373, 2.751700500332949, 3.302418177993942, 3.765426944214685, 4.244936269349019, 4.886737073814798, 5.473504133404292, 6.031565412744854, 6.196074168211381, 6.747504393408174, 6.892110854579573, 7.774920496346804, 8.448984103220546, 8.609485278189752, 9.334482213088615, 9.739957229709472, 10.26600596657329, 10.57654373846570, 11.00843535570219, 11.60178527287197, 11.89365269429954, 12.39710737743799, 12.80206643818519
