| L(s) = 1 | − 3-s + 5-s − 1.20·7-s + 9-s + 5.26·11-s + 0.221·13-s − 15-s − 4.28·17-s − 5.28·19-s + 1.20·21-s − 2.67·23-s + 25-s − 27-s − 6.52·29-s + 2.38·31-s − 5.26·33-s − 1.20·35-s + 8.84·37-s − 0.221·39-s − 7.02·41-s − 9.44·43-s + 45-s + 1.74·47-s − 5.54·49-s + 4.28·51-s − 5.05·53-s + 5.26·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.456·7-s + 0.333·9-s + 1.58·11-s + 0.0614·13-s − 0.258·15-s − 1.03·17-s − 1.21·19-s + 0.263·21-s − 0.558·23-s + 0.200·25-s − 0.192·27-s − 1.21·29-s + 0.428·31-s − 0.917·33-s − 0.204·35-s + 1.45·37-s − 0.0354·39-s − 1.09·41-s − 1.44·43-s + 0.149·45-s + 0.254·47-s − 0.791·49-s + 0.600·51-s − 0.694·53-s + 0.710·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 - 0.221T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 + 5.28T + 19T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 + 9.44T + 43T^{2} \) |
| 47 | \( 1 - 1.74T + 47T^{2} \) |
| 53 | \( 1 + 5.05T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 2.62T + 61T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.182536672084088520906101027250, −7.05756829902082507944695609165, −6.33594503056326534880884382349, −6.25598544790074460184141432562, −5.07744322067557247815078601241, −4.24338849498077599653239029906, −3.61565088194441640044875780858, −2.29097604605577277481109432376, −1.43365664974333987350102799626, 0,
1.43365664974333987350102799626, 2.29097604605577277481109432376, 3.61565088194441640044875780858, 4.24338849498077599653239029906, 5.07744322067557247815078601241, 6.25598544790074460184141432562, 6.33594503056326534880884382349, 7.05756829902082507944695609165, 8.182536672084088520906101027250
