| L(s) = 1 | − 1.15·3-s + 0.502·5-s − 3.23·7-s − 1.65·9-s + 3.32·11-s − 1.31·13-s − 0.582·15-s − 17-s − 1.69·19-s + 3.75·21-s − 1.10·23-s − 4.74·25-s + 5.39·27-s + 7.48·29-s + 8.21·31-s − 3.85·33-s − 1.62·35-s + 1.09·37-s + 1.52·39-s − 2.30·41-s − 10.4·43-s − 0.833·45-s − 8.64·47-s + 3.48·49-s + 1.15·51-s − 10.9·53-s + 1.67·55-s + ⋯ |
| L(s) = 1 | − 0.669·3-s + 0.224·5-s − 1.22·7-s − 0.552·9-s + 1.00·11-s − 0.365·13-s − 0.150·15-s − 0.242·17-s − 0.388·19-s + 0.818·21-s − 0.230·23-s − 0.949·25-s + 1.03·27-s + 1.38·29-s + 1.47·31-s − 0.670·33-s − 0.275·35-s + 0.179·37-s + 0.244·39-s − 0.359·41-s − 1.59·43-s − 0.124·45-s − 1.26·47-s + 0.497·49-s + 0.162·51-s − 1.50·53-s + 0.225·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8195538674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8195538674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 - 0.502T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 - 3.32T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 19 | \( 1 + 1.69T + 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 - 7.48T + 29T^{2} \) |
| 31 | \( 1 - 8.21T + 31T^{2} \) |
| 37 | \( 1 - 1.09T + 37T^{2} \) |
| 41 | \( 1 + 2.30T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 61 | \( 1 + 8.18T + 61T^{2} \) |
| 67 | \( 1 + 7.25T + 67T^{2} \) |
| 71 | \( 1 - 5.97T + 71T^{2} \) |
| 73 | \( 1 + 2.07T + 73T^{2} \) |
| 79 | \( 1 + 2.22T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−7.88300786576426588736603283126, −6.69234707002460597442460592600, −6.44004403213798286256633804240, −6.05606683104474600512234825161, −5.01707369163655194197728557000, −4.42339872573812973117224881140, −3.38695588327293639230309214396, −2.83204678707778445037170917758, −1.68779600832624594280034925469, −0.45544704984139759547807399763,
0.45544704984139759547807399763, 1.68779600832624594280034925469, 2.83204678707778445037170917758, 3.38695588327293639230309214396, 4.42339872573812973117224881140, 5.01707369163655194197728557000, 6.05606683104474600512234825161, 6.44004403213798286256633804240, 6.69234707002460597442460592600, 7.88300786576426588736603283126
