| L(s) = 1 | + 2.38·3-s − 1.68·5-s + 2.69·9-s − 4.39·11-s − 2.11·13-s − 4.01·15-s + 6.64·17-s + 3.59·19-s − 23-s − 2.17·25-s − 0.727·27-s + 9.27·29-s − 7.79·31-s − 10.4·33-s + 7.00·37-s − 5.05·39-s − 3.42·41-s − 5.89·43-s − 4.53·45-s − 8.85·47-s + 15.8·51-s + 4.27·53-s + 7.38·55-s + 8.57·57-s − 1.17·59-s − 1.43·61-s + 3.56·65-s + ⋯ |
| L(s) = 1 | + 1.37·3-s − 0.752·5-s + 0.898·9-s − 1.32·11-s − 0.587·13-s − 1.03·15-s + 1.61·17-s + 0.824·19-s − 0.208·23-s − 0.434·25-s − 0.140·27-s + 1.72·29-s − 1.40·31-s − 1.82·33-s + 1.15·37-s − 0.809·39-s − 0.534·41-s − 0.899·43-s − 0.675·45-s − 1.29·47-s + 2.21·51-s + 0.587·53-s + 0.995·55-s + 1.13·57-s − 0.152·59-s − 0.183·61-s + 0.441·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.38T + 3T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 19 | \( 1 - 3.59T + 19T^{2} \) |
| 29 | \( 1 - 9.27T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 - 7.00T + 37T^{2} \) |
| 41 | \( 1 + 3.42T + 41T^{2} \) |
| 43 | \( 1 + 5.89T + 43T^{2} \) |
| 47 | \( 1 + 8.85T + 47T^{2} \) |
| 53 | \( 1 - 4.27T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 + 1.43T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 7.15T + 71T^{2} \) |
| 73 | \( 1 + 2.35T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 6.68T + 83T^{2} \) |
| 89 | \( 1 - 1.10T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.52308220259950168889538347968, −7.20364969767557620660023427609, −5.93692269428055798388283007084, −5.22580916329555588180780324638, −4.50472865276433896592837705800, −3.53800191121068117319933139495, −3.10043043671150487368820157365, −2.47548842026156428353358252381, −1.39279920903496942526029100670, 0,
1.39279920903496942526029100670, 2.47548842026156428353358252381, 3.10043043671150487368820157365, 3.53800191121068117319933139495, 4.50472865276433896592837705800, 5.22580916329555588180780324638, 5.93692269428055798388283007084, 7.20364969767557620660023427609, 7.52308220259950168889538347968
