| L(s) = 1 | − 1.67·3-s − 5-s − 7-s − 0.193·9-s − 11-s − 6.63·13-s + 1.67·15-s + 4.63·17-s + 2·19-s + 1.67·21-s + 4.15·23-s + 25-s + 5.35·27-s − 7.92·29-s + 3.51·31-s + 1.67·33-s + 35-s + 4.54·37-s + 11.1·39-s + 2.48·41-s − 0.0303·43-s + 0.193·45-s + 12.9·47-s + 49-s − 7.76·51-s + 1.38·53-s + 55-s + ⋯ |
| L(s) = 1 | − 0.967·3-s − 0.447·5-s − 0.377·7-s − 0.0646·9-s − 0.301·11-s − 1.84·13-s + 0.432·15-s + 1.12·17-s + 0.458·19-s + 0.365·21-s + 0.866·23-s + 0.200·25-s + 1.02·27-s − 1.47·29-s + 0.631·31-s + 0.291·33-s + 0.169·35-s + 0.747·37-s + 1.78·39-s + 0.387·41-s − 0.00462·43-s + 0.0289·45-s + 1.88·47-s + 0.142·49-s − 1.08·51-s + 0.189·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 13 | \( 1 + 6.63T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4.15T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 3.51T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 - 2.48T + 41T^{2} \) |
| 43 | \( 1 + 0.0303T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 + 1.25T + 61T^{2} \) |
| 67 | \( 1 + 4.80T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 + 5.93T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 6.26T + 97T^{2} \) |
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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
−7.44633930252952543893834864073, −7.18578037943295379808931209070, −6.17046915046168175854062831171, −5.46694139146469000516069687293, −5.03738632211597582560428864197, −4.18239686722550355334126827475, −3.12732858519798277896011338199, −2.47765967950047161002073527160, −0.957776056858220043897549417083, 0,
0.957776056858220043897549417083, 2.47765967950047161002073527160, 3.12732858519798277896011338199, 4.18239686722550355334126827475, 5.03738632211597582560428864197, 5.46694139146469000516069687293, 6.17046915046168175854062831171, 7.18578037943295379808931209070, 7.44633930252952543893834864073
