| L(s) = 1 | − 2.07·3-s − 3.30·5-s + 0.766·7-s + 1.29·9-s − 4.04·11-s − 3.72·13-s + 6.84·15-s − 17-s + 0.733·19-s − 1.58·21-s − 1.80·23-s + 5.90·25-s + 3.52·27-s + 2.90·29-s + 1.23·31-s + 8.37·33-s − 2.53·35-s − 6.26·37-s + 7.72·39-s + 7.97·41-s + 0.613·43-s − 4.28·45-s + 13.3·47-s − 6.41·49-s + 2.07·51-s + 0.757·53-s + 13.3·55-s + ⋯ |
| L(s) = 1 | − 1.19·3-s − 1.47·5-s + 0.289·7-s + 0.432·9-s − 1.21·11-s − 1.03·13-s + 1.76·15-s − 0.242·17-s + 0.168·19-s − 0.346·21-s − 0.376·23-s + 1.18·25-s + 0.679·27-s + 0.540·29-s + 0.221·31-s + 1.45·33-s − 0.427·35-s − 1.03·37-s + 1.23·39-s + 1.24·41-s + 0.0935·43-s − 0.638·45-s + 1.95·47-s − 0.916·49-s + 0.290·51-s + 0.104·53-s + 1.79·55-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.07T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 7 | \( 1 - 0.766T + 7T^{2} \) |
| 11 | \( 1 + 4.04T + 11T^{2} \) |
| 13 | \( 1 + 3.72T + 13T^{2} \) |
| 19 | \( 1 - 0.733T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 + 6.26T + 37T^{2} \) |
| 41 | \( 1 - 7.97T + 41T^{2} \) |
| 43 | \( 1 - 0.613T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 0.757T + 53T^{2} \) |
| 61 | \( 1 - 4.13T + 61T^{2} \) |
| 67 | \( 1 + 9.80T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 - 9.46T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 + 0.845T + 83T^{2} \) |
| 89 | \( 1 + 0.654T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.42769949142947304651624615302, −6.95671108302790526018188472548, −5.99127204715588889308024149472, −5.29697210546240100400794239664, −4.73677788816731435618693973829, −4.17220598013541935553147983276, −3.13320505616159000675255968505, −2.30431281641749004316734334960, −0.75239854780050374802032284045, 0,
0.75239854780050374802032284045, 2.30431281641749004316734334960, 3.13320505616159000675255968505, 4.17220598013541935553147983276, 4.73677788816731435618693973829, 5.29697210546240100400794239664, 5.99127204715588889308024149472, 6.95671108302790526018188472548, 7.42769949142947304651624615302
