| L(s) = 1 | + 2.30·5-s + 2.60·7-s − 2.30·11-s + 1.30·13-s + 6·17-s − 2·19-s + 3.90·23-s + 0.302·25-s + 3.90·29-s + 0.302·31-s + 6·35-s + 37-s − 9.90·41-s − 0.605·43-s + 4.60·47-s − 0.211·49-s + 6·53-s − 5.30·55-s + 10.6·59-s + 7.51·61-s + 3·65-s + 3.51·67-s + 6·71-s − 12.3·73-s − 6·77-s − 9.11·79-s + 2.78·83-s + ⋯ |
| L(s) = 1 | + 1.02·5-s + 0.984·7-s − 0.694·11-s + 0.361·13-s + 1.45·17-s − 0.458·19-s + 0.814·23-s + 0.0605·25-s + 0.725·29-s + 0.0543·31-s + 1.01·35-s + 0.164·37-s − 1.54·41-s − 0.0923·43-s + 0.671·47-s − 0.0301·49-s + 0.824·53-s − 0.715·55-s + 1.38·59-s + 0.962·61-s + 0.372·65-s + 0.429·67-s + 0.712·71-s − 1.43·73-s − 0.683·77-s − 1.02·79-s + 0.306·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.952245650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952245650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 2.30T + 5T^{2} \) |
| 7 | \( 1 - 2.60T + 7T^{2} \) |
| 11 | \( 1 + 2.30T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.90T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 - 0.302T + 31T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 0.605T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 - 3.51T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 9.11T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.323592983242627874917213492577, −7.50372820445940605834998444674, −6.76351137251009962728426935201, −5.83497638758383679892330156399, −5.35167179157554105342310739398, −4.73968782034288584841232632340, −3.65389665183848135844145405089, −2.69480437711049096736712610966, −1.85861221642529668901453922658, −0.989011508325215839260393994108,
0.989011508325215839260393994108, 1.85861221642529668901453922658, 2.69480437711049096736712610966, 3.65389665183848135844145405089, 4.73968782034288584841232632340, 5.35167179157554105342310739398, 5.83497638758383679892330156399, 6.76351137251009962728426935201, 7.50372820445940605834998444674, 8.323592983242627874917213492577
