| L(s) = 1 | + 2.85·5-s − 7-s + 2.34·11-s − 0.859·13-s − 4.34·17-s − 19-s − 4.85·23-s + 3.17·25-s − 9.37·29-s − 7.37·31-s − 2.85·35-s + 2.17·37-s − 6.69·41-s + 0.353·43-s − 5.54·47-s + 49-s − 9.55·53-s + 6.69·55-s + 14.5·59-s − 12.9·61-s − 2.45·65-s − 4.34·67-s − 7.89·71-s + 2.23·73-s − 2.34·77-s + 7.36·79-s + 1.83·83-s + ⋯ |
| L(s) = 1 | + 1.27·5-s − 0.377·7-s + 0.705·11-s − 0.238·13-s − 1.05·17-s − 0.229·19-s − 1.01·23-s + 0.635·25-s − 1.74·29-s − 1.32·31-s − 0.483·35-s + 0.357·37-s − 1.04·41-s + 0.0539·43-s − 0.808·47-s + 0.142·49-s − 1.31·53-s + 0.902·55-s + 1.89·59-s − 1.65·61-s − 0.304·65-s − 0.530·67-s − 0.937·71-s + 0.261·73-s − 0.266·77-s + 0.828·79-s + 0.201·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.85T + 5T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 + 0.859T + 13T^{2} \) |
| 17 | \( 1 + 4.34T + 17T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 + 9.37T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 6.69T + 41T^{2} \) |
| 43 | \( 1 - 0.353T + 43T^{2} \) |
| 47 | \( 1 + 5.54T + 47T^{2} \) |
| 53 | \( 1 + 9.55T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 4.34T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 - 2.23T + 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 - 1.83T + 83T^{2} \) |
| 89 | \( 1 - 3.31T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.899385019936813032926048965988, −7.05525600287532077528601368695, −6.35892855877094674435060863559, −5.87936516085587305840767800565, −5.08894741655729006122253346542, −4.14156748431979660425516907669, −3.33193055776557401191519542690, −2.12635607869483977801507754761, −1.71142352880627929744876132350, 0,
1.71142352880627929744876132350, 2.12635607869483977801507754761, 3.33193055776557401191519542690, 4.14156748431979660425516907669, 5.08894741655729006122253346542, 5.87936516085587305840767800565, 6.35892855877094674435060863559, 7.05525600287532077528601368695, 7.899385019936813032926048965988
