| L(s) = 1 | − 1.41·3-s + 1.41·5-s + 7-s − 0.999·9-s + 2.82·11-s − 4.24·13-s − 2.00·15-s − 6·17-s + 4.24·19-s − 1.41·21-s − 6·23-s − 2.99·25-s + 5.65·27-s + 2.82·29-s + 4·31-s − 4.00·33-s + 1.41·35-s − 8.48·37-s + 6·39-s − 6·41-s − 8.48·43-s − 1.41·45-s + 49-s + 8.48·51-s + 5.65·53-s + 4.00·55-s − 6·57-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 0.632·5-s + 0.377·7-s − 0.333·9-s + 0.852·11-s − 1.17·13-s − 0.516·15-s − 1.45·17-s + 0.973·19-s − 0.308·21-s − 1.25·23-s − 0.599·25-s + 1.08·27-s + 0.525·29-s + 0.718·31-s − 0.696·33-s + 0.239·35-s − 1.39·37-s + 0.960·39-s − 0.937·41-s − 1.29·43-s − 0.210·45-s + 0.142·49-s + 1.18·51-s + 0.777·53-s + 0.539·55-s − 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.889346229009537588100192753073, −8.211053671223897493219664426993, −7.00288094530850823195406850826, −6.51389090631000240870412292670, −5.57108075871201719324647644270, −4.98550753293862563313877358269, −4.03470123020315719399587969642, −2.64811402327686523143820649711, −1.63152966807245294704758109688, 0,
1.63152966807245294704758109688, 2.64811402327686523143820649711, 4.03470123020315719399587969642, 4.98550753293862563313877358269, 5.57108075871201719324647644270, 6.51389090631000240870412292670, 7.00288094530850823195406850826, 8.211053671223897493219664426993, 8.889346229009537588100192753073
