| L(s) = 1 | − 2.56·3-s + 5-s + 1.56·7-s + 3.56·9-s − 2·11-s + 0.561·13-s − 2.56·15-s − 1.56·17-s − 6·19-s − 4·21-s − 23-s + 25-s − 1.43·27-s − 2.12·29-s + 9.24·31-s + 5.12·33-s + 1.56·35-s − 0.438·37-s − 1.43·39-s − 4.12·41-s + 3.56·45-s + 7.68·47-s − 4.56·49-s + 4·51-s − 0.438·53-s − 2·55-s + 15.3·57-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.447·5-s + 0.590·7-s + 1.18·9-s − 0.603·11-s + 0.155·13-s − 0.661·15-s − 0.378·17-s − 1.37·19-s − 0.872·21-s − 0.208·23-s + 0.200·25-s − 0.276·27-s − 0.394·29-s + 1.66·31-s + 0.891·33-s + 0.263·35-s − 0.0720·37-s − 0.230·39-s − 0.643·41-s + 0.530·45-s + 1.12·47-s − 0.651·49-s + 0.560·51-s − 0.0602·53-s − 0.269·55-s + 2.03·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 0.561T + 13T^{2} \) |
| 17 | \( 1 + 1.56T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 - 9.24T + 31T^{2} \) |
| 37 | \( 1 + 0.438T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 + 0.438T + 53T^{2} \) |
| 59 | \( 1 + 8.68T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 4.43T + 67T^{2} \) |
| 71 | \( 1 + 1.87T + 71T^{2} \) |
| 73 | \( 1 + 8.56T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−8.793849388000070594543696754177, −8.093801949440085159982244378357, −7.02656772355351363048196477379, −6.29133643802436696104142495865, −5.69588740487162721821107475148, −4.85738872382545566253811259373, −4.24864254885099928679391452767, −2.62532269527398296788564353162, −1.44146126821444297940418773491, 0,
1.44146126821444297940418773491, 2.62532269527398296788564353162, 4.24864254885099928679391452767, 4.85738872382545566253811259373, 5.69588740487162721821107475148, 6.29133643802436696104142495865, 7.02656772355351363048196477379, 8.093801949440085159982244378357, 8.793849388000070594543696754177
