| L(s) = 1 | + 3·3-s − 7.21·5-s + 9·9-s + 7.21·11-s + 13·13-s − 21.6·15-s + 26.9·25-s + 27·27-s + 21.6·33-s + 39·39-s + 79.3·41-s + 70·43-s − 64.8·45-s + 93.7·47-s + 49·49-s − 51.9·55-s + 7.21·59-s − 70·61-s − 93.7·65-s − 79.3·71-s + 80.9·75-s − 50·79-s + 81·81-s − 165.·83-s + 79.3·89-s + 64.8·99-s − 50·103-s + ⋯ |
| L(s) = 1 | + 3-s − 1.44·5-s + 9-s + 0.655·11-s + 13-s − 1.44·15-s + 1.07·25-s + 27-s + 0.655·33-s + 39-s + 1.93·41-s + 1.62·43-s − 1.44·45-s + 1.99·47-s + 0.999·49-s − 0.945·55-s + 0.122·59-s − 1.14·61-s − 1.44·65-s − 1.11·71-s + 1.07·75-s − 0.632·79-s + 81-s − 1.99·83-s + 0.891·89-s + 0.655·99-s − 0.485·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.217227622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.217227622\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 + 7.21T + 25T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 - 7.21T + 121T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 79.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 70T + 1.84e3T^{2} \) |
| 47 | \( 1 - 93.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 7.21T + 3.48e3T^{2} \) |
| 61 | \( 1 + 70T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 + 79.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 + 50T + 6.24e3T^{2} \) |
| 83 | \( 1 + 165.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 79.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−10.48971907145748870050973484116, −9.168474206967900581099621028066, −8.724847779351188150133352929595, −7.73120417651566423161181118449, −7.24057857847381548701244232051, −5.98692488931039045107906146072, −4.24531286914222226586461521236, −3.91638444687498764378212914362, −2.73814021218436884015473866435, −1.04763083600351915015704116253,
1.04763083600351915015704116253, 2.73814021218436884015473866435, 3.91638444687498764378212914362, 4.24531286914222226586461521236, 5.98692488931039045107906146072, 7.24057857847381548701244232051, 7.73120417651566423161181118449, 8.724847779351188150133352929595, 9.168474206967900581099621028066, 10.48971907145748870050973484116
