| L(s) = 1 | − 3.18·3-s − 0.628·5-s + 2.91·7-s + 7.11·9-s − 1.79·11-s − 0.925·13-s + 1.99·15-s − 2.99·17-s − 2.66·19-s − 9.28·21-s + 5.21·23-s − 4.60·25-s − 13.0·27-s − 6.23·29-s + 8.37·31-s + 5.71·33-s − 1.83·35-s − 10.3·37-s + 2.94·39-s + 7.74·41-s + 12.2·43-s − 4.47·45-s − 2.78·47-s + 1.51·49-s + 9.51·51-s + 8.90·53-s + 1.12·55-s + ⋯ |
| L(s) = 1 | − 1.83·3-s − 0.281·5-s + 1.10·7-s + 2.37·9-s − 0.541·11-s − 0.256·13-s + 0.516·15-s − 0.725·17-s − 0.610·19-s − 2.02·21-s + 1.08·23-s − 0.920·25-s − 2.52·27-s − 1.15·29-s + 1.50·31-s + 0.995·33-s − 0.309·35-s − 1.70·37-s + 0.471·39-s + 1.20·41-s + 1.86·43-s − 0.666·45-s − 0.406·47-s + 0.215·49-s + 1.33·51-s + 1.22·53-s + 0.152·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7690243189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7690243189\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 + 0.628T + 5T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 + 1.79T + 11T^{2} \) |
| 13 | \( 1 + 0.925T + 13T^{2} \) |
| 17 | \( 1 + 2.99T + 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 - 5.21T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 8.37T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.74T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 2.78T + 47T^{2} \) |
| 53 | \( 1 - 8.90T + 53T^{2} \) |
| 59 | \( 1 + 1.47T + 59T^{2} \) |
| 61 | \( 1 + 0.606T + 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 5.84T + 79T^{2} \) |
| 83 | \( 1 - 9.83T + 83T^{2} \) |
| 89 | \( 1 - 1.16T + 89T^{2} \) |
| 97 | \( 1 - 1.30T + 97T^{2} \) |
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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
−7.50259345157012915518289634541, −7.21222844645766762501707508417, −6.28529616179427165108665747239, −5.71530710813289217070483344702, −5.03031404782219070995766257376, −4.56069711806236514168492720393, −3.91923540520567362533375966775, −2.44129920134163763929191819422, −1.51665394074562731690105168143, −0.49849656104486264303894915786,
0.49849656104486264303894915786, 1.51665394074562731690105168143, 2.44129920134163763929191819422, 3.91923540520567362533375966775, 4.56069711806236514168492720393, 5.03031404782219070995766257376, 5.71530710813289217070483344702, 6.28529616179427165108665747239, 7.21222844645766762501707508417, 7.50259345157012915518289634541
