| L(s) = 1 | − 1.38·3-s − 2.65·5-s − 3.15·7-s − 1.08·9-s + 2.64·11-s + 3.71·13-s + 3.66·15-s + 1.33·17-s − 8.19·19-s + 4.36·21-s − 2.10·23-s + 2.02·25-s + 5.65·27-s − 9.86·29-s − 1.00·31-s − 3.65·33-s + 8.36·35-s + 11.4·37-s − 5.12·39-s + 8.32·41-s + 3.38·43-s + 2.88·45-s + 3.23·47-s + 2.96·49-s − 1.84·51-s + 10.1·53-s − 7.01·55-s + ⋯ |
| L(s) = 1 | − 0.797·3-s − 1.18·5-s − 1.19·7-s − 0.363·9-s + 0.797·11-s + 1.02·13-s + 0.946·15-s + 0.323·17-s − 1.88·19-s + 0.951·21-s − 0.438·23-s + 0.405·25-s + 1.08·27-s − 1.83·29-s − 0.179·31-s − 0.636·33-s + 1.41·35-s + 1.88·37-s − 0.821·39-s + 1.29·41-s + 0.516·43-s + 0.430·45-s + 0.471·47-s + 0.423·49-s − 0.258·51-s + 1.38·53-s − 0.945·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 1.38T + 3T^{2} \) |
| 5 | \( 1 + 2.65T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 + 8.19T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 + 9.86T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 9.28T + 59T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 - 6.35T + 73T^{2} \) |
| 79 | \( 1 - 2.00T + 79T^{2} \) |
| 83 | \( 1 + 2.85T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 - 7.40T + 97T^{2} \) |
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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
−7.46631321656901917258029533221, −6.59865081402016763498500781567, −6.08295746413079233490005885221, −5.74153581391330649964757863791, −4.38483322561471653897307627020, −3.97102675202074065261144738276, −3.36426364345588339344110407054, −2.28328131654447854601219604814, −0.838184135896557358267290234154, 0,
0.838184135896557358267290234154, 2.28328131654447854601219604814, 3.36426364345588339344110407054, 3.97102675202074065261144738276, 4.38483322561471653897307627020, 5.74153581391330649964757863791, 6.08295746413079233490005885221, 6.59865081402016763498500781567, 7.46631321656901917258029533221
