| L(s) = 1 | − 1.57·2-s + 0.480·4-s − 0.664·5-s − 4.15·7-s + 2.39·8-s + 1.04·10-s + 11-s + 2.20·13-s + 6.53·14-s − 4.73·16-s + 0.555·17-s − 6.78·19-s − 0.319·20-s − 1.57·22-s + 4.88·23-s − 4.55·25-s − 3.47·26-s − 1.99·28-s − 6.82·29-s − 4.34·31-s + 2.66·32-s − 0.874·34-s + 2.76·35-s − 5.04·37-s + 10.6·38-s − 1.59·40-s + 2.62·41-s + ⋯ |
| L(s) = 1 | − 1.11·2-s + 0.240·4-s − 0.297·5-s − 1.56·7-s + 0.846·8-s + 0.331·10-s + 0.301·11-s + 0.612·13-s + 1.74·14-s − 1.18·16-s + 0.134·17-s − 1.55·19-s − 0.0714·20-s − 0.335·22-s + 1.01·23-s − 0.911·25-s − 0.681·26-s − 0.377·28-s − 1.26·29-s − 0.781·31-s + 0.471·32-s − 0.149·34-s + 0.466·35-s − 0.829·37-s + 1.73·38-s − 0.251·40-s + 0.410·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3514166506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3514166506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.57T + 2T^{2} \) |
| 5 | \( 1 + 0.664T + 5T^{2} \) |
| 7 | \( 1 + 4.15T + 7T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 - 0.555T + 17T^{2} \) |
| 19 | \( 1 + 6.78T + 19T^{2} \) |
| 23 | \( 1 - 4.88T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + 4.34T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 2.62T + 41T^{2} \) |
| 43 | \( 1 - 7.39T + 43T^{2} \) |
| 47 | \( 1 + 8.12T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 67 | \( 1 + 0.106T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 3.84T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 8.42T + 97T^{2} \) |
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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
−8.337635174421118566157635856474, −7.30905462638908845468548851514, −6.94180457640411526545307865141, −6.13887318033917166777205183431, −5.39142523001417814687407109004, −4.06371715334397369756578440879, −3.80338091626190471100609950694, −2.64466401123209939142096158587, −1.59088590934624728805465097870, −0.37214331708225256792925185649,
0.37214331708225256792925185649, 1.59088590934624728805465097870, 2.64466401123209939142096158587, 3.80338091626190471100609950694, 4.06371715334397369756578440879, 5.39142523001417814687407109004, 6.13887318033917166777205183431, 6.94180457640411526545307865141, 7.30905462638908845468548851514, 8.337635174421118566157635856474
