| L(s) = 1 | − 1.35·3-s + 5-s − 0.648·7-s − 1.17·9-s − 3.35·11-s − 4.17·13-s − 1.35·15-s + 4.82·17-s − 6.82·19-s + 0.876·21-s + 5.52·23-s + 25-s + 5.64·27-s + 29-s − 2.82·31-s + 4.53·33-s − 0.648·35-s + 10.2·37-s + 5.64·39-s + 8.17·41-s + 5.69·43-s − 1.17·45-s − 2.64·47-s − 6.58·49-s − 6.51·51-s + 2.87·53-s − 3.35·55-s + ⋯ |
| L(s) = 1 | − 0.780·3-s + 0.447·5-s − 0.244·7-s − 0.390·9-s − 1.01·11-s − 1.15·13-s − 0.349·15-s + 1.16·17-s − 1.56·19-s + 0.191·21-s + 1.15·23-s + 0.200·25-s + 1.08·27-s + 0.185·29-s − 0.506·31-s + 0.788·33-s − 0.109·35-s + 1.68·37-s + 0.903·39-s + 1.27·41-s + 0.868·43-s − 0.174·45-s − 0.386·47-s − 0.940·49-s − 0.912·51-s + 0.395·53-s − 0.451·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.35T + 3T^{2} \) |
| 7 | \( 1 + 0.648T + 7T^{2} \) |
| 11 | \( 1 + 3.35T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 - 8.17T + 41T^{2} \) |
| 43 | \( 1 - 5.69T + 43T^{2} \) |
| 47 | \( 1 + 2.64T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 8.87T + 71T^{2} \) |
| 73 | \( 1 - 9.69T + 73T^{2} \) |
| 79 | \( 1 - 8.99T + 79T^{2} \) |
| 83 | \( 1 + 1.94T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−7.36474845551844160442939395196, −6.52184939675738630649343147919, −5.98705323828913852190704659362, −5.26531430905038674971687544940, −4.91043433286482236565443491363, −3.92986168420535948073313371013, −2.70424092354768665629015235224, −2.47429003784572484998331690856, −0.999637294841537897974002833620, 0,
0.999637294841537897974002833620, 2.47429003784572484998331690856, 2.70424092354768665629015235224, 3.92986168420535948073313371013, 4.91043433286482236565443491363, 5.26531430905038674971687544940, 5.98705323828913852190704659362, 6.52184939675738630649343147919, 7.36474845551844160442939395196
