| L(s) = 1 | − 3-s − 2.21·5-s − 4.16·7-s + 9-s + 6.39·11-s + 2.21·15-s − 6.12·17-s − 2.16·19-s + 4.16·21-s − 0.597·23-s − 0.115·25-s − 27-s − 3.01·29-s − 8.39·31-s − 6.39·33-s + 9.20·35-s − 6.61·37-s + 6.03·41-s + 11.6·43-s − 2.21·45-s − 8.31·47-s + 10.3·49-s + 6.12·51-s + 7.73·53-s − 14.1·55-s + 2.16·57-s + 0.123·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.988·5-s − 1.57·7-s + 0.333·9-s + 1.92·11-s + 0.570·15-s − 1.48·17-s − 0.495·19-s + 0.908·21-s − 0.124·23-s − 0.0230·25-s − 0.192·27-s − 0.559·29-s − 1.50·31-s − 1.11·33-s + 1.55·35-s − 1.08·37-s + 0.943·41-s + 1.78·43-s − 0.329·45-s − 1.21·47-s + 1.47·49-s + 0.857·51-s + 1.06·53-s − 1.90·55-s + 0.286·57-s + 0.0161·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5675220358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5675220358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 + 4.16T + 7T^{2} \) |
| 11 | \( 1 - 6.39T + 11T^{2} \) |
| 17 | \( 1 + 6.12T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 + 0.597T + 23T^{2} \) |
| 29 | \( 1 + 3.01T + 29T^{2} \) |
| 31 | \( 1 + 8.39T + 31T^{2} \) |
| 37 | \( 1 + 6.61T + 37T^{2} \) |
| 41 | \( 1 - 6.03T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 - 0.123T + 59T^{2} \) |
| 61 | \( 1 - 7.22T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 + 5.78T + 71T^{2} \) |
| 73 | \( 1 + 0.175T + 73T^{2} \) |
| 79 | \( 1 + 2.74T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 1.65T + 97T^{2} \) |
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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
−8.712205586488554836500229292650, −7.34856069712123815910870672569, −6.97216395881441161940051440266, −6.30156749447178342552346086237, −5.72265978144390191396025497005, −4.22830011327855479466154840733, −4.07219224059736719577244558478, −3.21491490804727380514721836162, −1.87100955788743314605237536715, −0.43033038892877802388904330222,
0.43033038892877802388904330222, 1.87100955788743314605237536715, 3.21491490804727380514721836162, 4.07219224059736719577244558478, 4.22830011327855479466154840733, 5.72265978144390191396025497005, 6.30156749447178342552346086237, 6.97216395881441161940051440266, 7.34856069712123815910870672569, 8.712205586488554836500229292650
