| L(s) = 1 | + 5-s + 7-s + 0.828·11-s − 4.82·13-s − 3.65·17-s − 2.82·19-s − 7.65·23-s + 25-s + 8·29-s + 8.48·31-s + 35-s − 6·37-s + 3.65·41-s + 9.65·43-s + 4·47-s + 49-s + 10.8·53-s + 0.828·55-s + 4·59-s + 6·61-s − 4.82·65-s − 7.31·67-s + 6.48·71-s + 16.1·73-s + 0.828·77-s + 5.65·79-s + 8·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.249·11-s − 1.33·13-s − 0.886·17-s − 0.648·19-s − 1.59·23-s + 0.200·25-s + 1.48·29-s + 1.52·31-s + 0.169·35-s − 0.986·37-s + 0.571·41-s + 1.47·43-s + 0.583·47-s + 0.142·49-s + 1.48·53-s + 0.111·55-s + 0.520·59-s + 0.768·61-s − 0.598·65-s − 0.893·67-s + 0.769·71-s + 1.88·73-s + 0.0944·77-s + 0.636·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.925717543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.925717543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 7.65T + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 8.48T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 7.31T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 5.65T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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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
−8.252527843099672409563395340507, −7.55883233015291440340340759909, −6.68261770715445771702424352774, −6.20691423380235556160963735016, −5.23599224703829190133863935792, −4.56575531990326388736346876230, −3.90675847689838815535100962389, −2.47807732860145028732357999379, −2.19581966552225479196198511978, −0.74061224418902330417568980671,
0.74061224418902330417568980671, 2.19581966552225479196198511978, 2.47807732860145028732357999379, 3.90675847689838815535100962389, 4.56575531990326388736346876230, 5.23599224703829190133863935792, 6.20691423380235556160963735016, 6.68261770715445771702424352774, 7.55883233015291440340340759909, 8.252527843099672409563395340507
