| L(s) = 1 | − 2.61·3-s − 0.436·5-s + 3.83·9-s + 11-s − 5.35·13-s + 1.14·15-s − 7.06·17-s − 8.05·19-s + 6.66·23-s − 4.80·25-s − 2.19·27-s − 4.59·29-s − 3.04·31-s − 2.61·33-s − 9.94·37-s + 13.9·39-s − 6.37·41-s − 11.2·43-s − 1.67·45-s + 8.72·47-s + 18.4·51-s + 4.97·53-s − 0.436·55-s + 21.0·57-s + 2.27·59-s + 3.55·61-s + 2.33·65-s + ⋯ |
| L(s) = 1 | − 1.50·3-s − 0.195·5-s + 1.27·9-s + 0.301·11-s − 1.48·13-s + 0.294·15-s − 1.71·17-s − 1.84·19-s + 1.39·23-s − 0.961·25-s − 0.422·27-s − 0.853·29-s − 0.546·31-s − 0.455·33-s − 1.63·37-s + 2.24·39-s − 0.996·41-s − 1.71·43-s − 0.249·45-s + 1.27·47-s + 2.58·51-s + 0.682·53-s − 0.0588·55-s + 2.79·57-s + 0.295·59-s + 0.454·61-s + 0.289·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04100805669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04100805669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 0.436T + 5T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 19 | \( 1 + 8.05T + 19T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + 3.04T + 31T^{2} \) |
| 37 | \( 1 + 9.94T + 37T^{2} \) |
| 41 | \( 1 + 6.37T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 - 2.27T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 9.31T + 67T^{2} \) |
| 71 | \( 1 - 0.846T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 - 8.05T + 79T^{2} \) |
| 83 | \( 1 + 9.08T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 7.77T + 97T^{2} \) |
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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
−7.43431031012289382874392009235, −6.86008875312706609776446730947, −6.55993955697415984362234920822, −5.63040656895365576286654640965, −5.01199467336545205530861947559, −4.49517787440206285673643950336, −3.74929872518349244441711979877, −2.43343102102382661101464912375, −1.69391894952305925177076416374, −0.10251559288994634238880646896,
0.10251559288994634238880646896, 1.69391894952305925177076416374, 2.43343102102382661101464912375, 3.74929872518349244441711979877, 4.49517787440206285673643950336, 5.01199467336545205530861947559, 5.63040656895365576286654640965, 6.55993955697415984362234920822, 6.86008875312706609776446730947, 7.43431031012289382874392009235
