| L(s) = 1 | − 2-s + 2.20·3-s + 4-s − 4.27·5-s − 2.20·6-s + 3.25·7-s − 8-s + 1.84·9-s + 4.27·10-s + 0.599·11-s + 2.20·12-s + 6.18·13-s − 3.25·14-s − 9.41·15-s + 16-s − 0.284·17-s − 1.84·18-s − 6.17·19-s − 4.27·20-s + 7.16·21-s − 0.599·22-s + 23-s − 2.20·24-s + 13.2·25-s − 6.18·26-s − 2.53·27-s + 3.25·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.27·3-s + 0.5·4-s − 1.91·5-s − 0.898·6-s + 1.22·7-s − 0.353·8-s + 0.616·9-s + 1.35·10-s + 0.180·11-s + 0.635·12-s + 1.71·13-s − 0.869·14-s − 2.43·15-s + 0.250·16-s − 0.0691·17-s − 0.435·18-s − 1.41·19-s − 0.955·20-s + 1.56·21-s − 0.127·22-s + 0.208·23-s − 0.449·24-s + 2.65·25-s − 1.21·26-s − 0.487·27-s + 0.614·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.913812845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.913812845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 3 | \( 1 - 2.20T + 3T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 - 0.599T + 11T^{2} \) |
| 13 | \( 1 - 6.18T + 13T^{2} \) |
| 17 | \( 1 + 0.284T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 29 | \( 1 - 6.36T + 29T^{2} \) |
| 31 | \( 1 - 2.02T + 31T^{2} \) |
| 37 | \( 1 - 4.31T + 37T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 0.750T + 47T^{2} \) |
| 53 | \( 1 + 1.34T + 53T^{2} \) |
| 59 | \( 1 - 2.09T + 59T^{2} \) |
| 61 | \( 1 - 0.425T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 + 0.803T + 79T^{2} \) |
| 83 | \( 1 - 8.19T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 9.54T + 97T^{2} \) |
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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
−8.208301838708358501649423373101, −7.85157186135484198395718078534, −6.97716728663924859284060947415, −6.29389533956297229700517209618, −4.90511947601793499967390301137, −4.13406935013543543255462759465, −3.63964074655256948351263950308, −2.82648138076615453031524844710, −1.77579686042663352052375349100, −0.78680760171640085687030395918,
0.78680760171640085687030395918, 1.77579686042663352052375349100, 2.82648138076615453031524844710, 3.63964074655256948351263950308, 4.13406935013543543255462759465, 4.90511947601793499967390301137, 6.29389533956297229700517209618, 6.97716728663924859284060947415, 7.85157186135484198395718078534, 8.208301838708358501649423373101
