| L(s) = 1 | + 1.19·3-s + 3.27·5-s − 4.85·7-s − 1.58·9-s − 3.40·11-s + 13-s + 3.90·15-s + 7.52·17-s − 0.394·19-s − 5.78·21-s − 1.27·23-s + 5.72·25-s − 5.45·27-s + 29-s + 6.58·31-s − 4.05·33-s − 15.9·35-s − 6.44·37-s + 1.19·39-s − 6.08·41-s − 12.4·43-s − 5.17·45-s − 3.24·47-s + 16.6·49-s + 8.96·51-s − 8.77·53-s − 11.1·55-s + ⋯ |
| L(s) = 1 | + 0.687·3-s + 1.46·5-s − 1.83·7-s − 0.527·9-s − 1.02·11-s + 0.277·13-s + 1.00·15-s + 1.82·17-s − 0.0905·19-s − 1.26·21-s − 0.266·23-s + 1.14·25-s − 1.05·27-s + 0.185·29-s + 1.18·31-s − 0.705·33-s − 2.68·35-s − 1.05·37-s + 0.190·39-s − 0.949·41-s − 1.90·43-s − 0.771·45-s − 0.472·47-s + 2.37·49-s + 1.25·51-s − 1.20·53-s − 1.50·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 + 4.85T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 17 | \( 1 - 7.52T + 17T^{2} \) |
| 19 | \( 1 + 0.394T + 19T^{2} \) |
| 23 | \( 1 + 1.27T + 23T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 + 6.44T + 37T^{2} \) |
| 41 | \( 1 + 6.08T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 + 6.06T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 - 7.89T + 71T^{2} \) |
| 73 | \( 1 + 3.23T + 73T^{2} \) |
| 79 | \( 1 - 3.97T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 7.15T + 97T^{2} \) |
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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
−7.922852313659994720761667325433, −6.78258610438952809169824650172, −6.33831019380432932513111044991, −5.58621750589524792262057879468, −5.17833307896340906369411222543, −3.62035962363877159756208892802, −3.06595238618490123039656221687, −2.59792359795614183816045409658, −1.51016737819915345317252164707, 0,
1.51016737819915345317252164707, 2.59792359795614183816045409658, 3.06595238618490123039656221687, 3.62035962363877159756208892802, 5.17833307896340906369411222543, 5.58621750589524792262057879468, 6.33831019380432932513111044991, 6.78258610438952809169824650172, 7.922852313659994720761667325433
