| L(s) = 1 | − 0.414·2-s + 3-s − 1.82·4-s + 0.585·5-s − 0.414·6-s + 4.82·7-s + 1.58·8-s + 9-s − 0.242·10-s − 11-s − 1.82·12-s − 1.99·14-s + 0.585·15-s + 3·16-s − 5.41·17-s − 0.414·18-s + 6·19-s − 1.07·20-s + 4.82·21-s + 0.414·22-s − 0.828·23-s + 1.58·24-s − 4.65·25-s + 27-s − 8.82·28-s + 4.24·29-s − 0.242·30-s + ⋯ |
| L(s) = 1 | − 0.292·2-s + 0.577·3-s − 0.914·4-s + 0.261·5-s − 0.169·6-s + 1.82·7-s + 0.560·8-s + 0.333·9-s − 0.0767·10-s − 0.301·11-s − 0.527·12-s − 0.534·14-s + 0.151·15-s + 0.750·16-s − 1.31·17-s − 0.0976·18-s + 1.37·19-s − 0.239·20-s + 1.05·21-s + 0.0883·22-s − 0.172·23-s + 0.323·24-s − 0.931·25-s + 0.192·27-s − 1.66·28-s + 0.787·29-s − 0.0442·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5577 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287860435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287860435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 17 | \( 1 + 5.41T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 3.07T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 9.07T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.240171532270910902197082306435, −7.59215719608768516288378307504, −7.13458293760398565929322791857, −5.63286853452267978851630179214, −5.29418664745968666441016094545, −4.33036299136918145589538040277, −4.00175633611168118667344816771, −2.60514067535432031069148921624, −1.81331509865071576210558127158, −0.876511622577928938912522752088,
0.876511622577928938912522752088, 1.81331509865071576210558127158, 2.60514067535432031069148921624, 4.00175633611168118667344816771, 4.33036299136918145589538040277, 5.29418664745968666441016094545, 5.63286853452267978851630179214, 7.13458293760398565929322791857, 7.59215719608768516288378307504, 8.240171532270910902197082306435
