| L(s) = 1 | − 2.79·5-s − 7-s − 1.36·11-s + 0.713·13-s + 7.33·17-s − 19-s − 8.34·23-s + 2.81·25-s + 6.15·29-s − 10.0·31-s + 2.79·35-s + 6.43·37-s + 7.98·41-s − 0.819·43-s − 6.54·47-s + 49-s − 7.88·53-s + 3.81·55-s + 0.796·59-s − 7.74·61-s − 1.99·65-s + 1.72·67-s − 2.18·71-s − 8.41·73-s + 1.36·77-s − 12.0·79-s − 11.8·83-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 0.377·7-s − 0.411·11-s + 0.197·13-s + 1.77·17-s − 0.229·19-s − 1.74·23-s + 0.563·25-s + 1.14·29-s − 1.80·31-s + 0.472·35-s + 1.05·37-s + 1.24·41-s − 0.124·43-s − 0.955·47-s + 0.142·49-s − 1.08·53-s + 0.514·55-s + 0.103·59-s − 0.991·61-s − 0.247·65-s + 0.210·67-s − 0.258·71-s − 0.984·73-s + 0.155·77-s − 1.35·79-s − 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9692878622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9692878622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.79T + 5T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 - 0.713T + 13T^{2} \) |
| 17 | \( 1 - 7.33T + 17T^{2} \) |
| 23 | \( 1 + 8.34T + 23T^{2} \) |
| 29 | \( 1 - 6.15T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 6.43T + 37T^{2} \) |
| 41 | \( 1 - 7.98T + 41T^{2} \) |
| 43 | \( 1 + 0.819T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 + 7.88T + 53T^{2} \) |
| 59 | \( 1 - 0.796T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 - 1.72T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 + 8.41T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 1.60T + 97T^{2} \) |
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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
−7.64885438192768573754832150921, −7.33271344086805242826127783666, −6.15785068947063311294224049635, −5.82057395370272413192759721615, −4.80336639073122746575084803044, −4.09556011056200710636129538343, −3.48130611320669319986202737805, −2.84219621854171545623799704923, −1.64318681807272342816068349509, −0.46993216460796959612503583020,
0.46993216460796959612503583020, 1.64318681807272342816068349509, 2.84219621854171545623799704923, 3.48130611320669319986202737805, 4.09556011056200710636129538343, 4.80336639073122746575084803044, 5.82057395370272413192759721615, 6.15785068947063311294224049635, 7.33271344086805242826127783666, 7.64885438192768573754832150921
