| L(s) = 1 | − 3-s − 2.48·5-s + 1.43·7-s + 9-s − 0.681·11-s − 1.52·13-s + 2.48·15-s − 0.279·17-s + 1.75·19-s − 1.43·21-s − 1.39·23-s + 1.15·25-s − 27-s + 9.33·29-s − 10.3·31-s + 0.681·33-s − 3.56·35-s + 3.71·37-s + 1.52·39-s + 6.04·41-s + 0.878·43-s − 2.48·45-s + 0.0496·47-s − 4.94·49-s + 0.279·51-s + 1.77·53-s + 1.69·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.10·5-s + 0.542·7-s + 0.333·9-s − 0.205·11-s − 0.423·13-s + 0.640·15-s − 0.0676·17-s + 0.401·19-s − 0.313·21-s − 0.290·23-s + 0.231·25-s − 0.192·27-s + 1.73·29-s − 1.85·31-s + 0.118·33-s − 0.602·35-s + 0.610·37-s + 0.244·39-s + 0.943·41-s + 0.133·43-s − 0.369·45-s + 0.00723·47-s − 0.705·49-s + 0.0390·51-s + 0.243·53-s + 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 173 | \( 1 - T \) |
| good | 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 + 0.681T + 11T^{2} \) |
| 13 | \( 1 + 1.52T + 13T^{2} \) |
| 17 | \( 1 + 0.279T + 17T^{2} \) |
| 19 | \( 1 - 1.75T + 19T^{2} \) |
| 23 | \( 1 + 1.39T + 23T^{2} \) |
| 29 | \( 1 - 9.33T + 29T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 - 3.71T + 37T^{2} \) |
| 41 | \( 1 - 6.04T + 41T^{2} \) |
| 43 | \( 1 - 0.878T + 43T^{2} \) |
| 47 | \( 1 - 0.0496T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 - 9.90T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 2.64T + 71T^{2} \) |
| 73 | \( 1 + 8.14T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 7.87T + 97T^{2} \) |
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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
−7.48324541262904007638037813632, −6.92181819309675036008620488588, −6.03165824380016865870989614452, −5.29887472041611555575095537153, −4.61668141628839114495452203353, −4.05108784471090909444180948831, −3.19714571465113452700519061613, −2.18961421069029621957547680129, −1.04355405024819523639415327628, 0,
1.04355405024819523639415327628, 2.18961421069029621957547680129, 3.19714571465113452700519061613, 4.05108784471090909444180948831, 4.61668141628839114495452203353, 5.29887472041611555575095537153, 6.03165824380016865870989614452, 6.92181819309675036008620488588, 7.48324541262904007638037813632
