| L(s) = 1 | − 0.410·3-s − 4.21·5-s − 3.36·7-s − 2.83·9-s − 4.13·11-s − 3.22·13-s + 1.72·15-s + 4.83·17-s + 7.13·19-s + 1.37·21-s + 0.264·23-s + 12.7·25-s + 2.39·27-s − 6.92·29-s − 1.07·31-s + 1.69·33-s + 14.1·35-s + 5.79·37-s + 1.32·39-s + 1.24·41-s − 3.97·43-s + 11.9·45-s + 7.96·47-s + 4.29·49-s − 1.98·51-s + 13.9·53-s + 17.4·55-s + ⋯ |
| L(s) = 1 | − 0.236·3-s − 1.88·5-s − 1.27·7-s − 0.943·9-s − 1.24·11-s − 0.894·13-s + 0.446·15-s + 1.17·17-s + 1.63·19-s + 0.300·21-s + 0.0551·23-s + 2.54·25-s + 0.460·27-s − 1.28·29-s − 0.192·31-s + 0.295·33-s + 2.39·35-s + 0.953·37-s + 0.211·39-s + 0.195·41-s − 0.606·43-s + 1.77·45-s + 1.16·47-s + 0.613·49-s − 0.277·51-s + 1.92·53-s + 2.35·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 + 0.410T + 3T^{2} \) |
| 5 | \( 1 + 4.21T + 5T^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 11 | \( 1 + 4.13T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 - 0.264T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.07T + 31T^{2} \) |
| 37 | \( 1 - 5.79T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 - 7.96T + 47T^{2} \) |
| 53 | \( 1 - 13.9T + 53T^{2} \) |
| 59 | \( 1 + 1.88T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 2.49T + 73T^{2} \) |
| 79 | \( 1 - 8.07T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.18T + 89T^{2} \) |
| 97 | \( 1 - 7.00T + 97T^{2} \) |
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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
−7.61635501677659891948245146339, −7.11560263066036381887860196053, −5.96042332619471606575024964760, −5.41608287173587378444150060301, −4.70666869614628745932775848572, −3.67303087259700448847863737747, −3.15872884298790283842760161050, −2.68532045063292646628763961708, −0.72335769595372058004493275778, 0,
0.72335769595372058004493275778, 2.68532045063292646628763961708, 3.15872884298790283842760161050, 3.67303087259700448847863737747, 4.70666869614628745932775848572, 5.41608287173587378444150060301, 5.96042332619471606575024964760, 7.11560263066036381887860196053, 7.61635501677659891948245146339
