| L(s) = 1 | − 2-s − 2.81·3-s + 4-s − 2.87·5-s + 2.81·6-s − 0.0915·7-s − 8-s + 4.90·9-s + 2.87·10-s − 11-s − 2.81·12-s + 6.84·13-s + 0.0915·14-s + 8.08·15-s + 16-s + 3.48·17-s − 4.90·18-s − 5.03·19-s − 2.87·20-s + 0.257·21-s + 22-s + 6.36·23-s + 2.81·24-s + 3.27·25-s − 6.84·26-s − 5.35·27-s − 0.0915·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.62·3-s + 0.5·4-s − 1.28·5-s + 1.14·6-s − 0.0346·7-s − 0.353·8-s + 1.63·9-s + 0.909·10-s − 0.301·11-s − 0.811·12-s + 1.89·13-s + 0.0244·14-s + 2.08·15-s + 0.250·16-s + 0.844·17-s − 1.15·18-s − 1.15·19-s − 0.643·20-s + 0.0561·21-s + 0.213·22-s + 1.32·23-s + 0.573·24-s + 0.654·25-s − 1.34·26-s − 1.03·27-s − 0.0173·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5498922558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5498922558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 2.81T + 3T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + 0.0915T + 7T^{2} \) |
| 13 | \( 1 - 6.84T + 13T^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 + 5.03T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 7.30T + 41T^{2} \) |
| 43 | \( 1 - 4.53T + 43T^{2} \) |
| 47 | \( 1 + 2.86T + 47T^{2} \) |
| 53 | \( 1 + 4.85T + 53T^{2} \) |
| 59 | \( 1 + 0.0471T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 7.84T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 9.55T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.226115314183148058787558262461, −7.70973661134228023047860327015, −6.87423630022792769741963754072, −6.18296583426565567876579612891, −5.70921861069314937927387256604, −4.59978031787177643735560559495, −3.99128467048986778068858671889, −2.98271453412759566863971876803, −1.29432159158094827290643464390, −0.58433117582755017643726114149,
0.58433117582755017643726114149, 1.29432159158094827290643464390, 2.98271453412759566863971876803, 3.99128467048986778068858671889, 4.59978031787177643735560559495, 5.70921861069314937927387256604, 6.18296583426565567876579612891, 6.87423630022792769741963754072, 7.70973661134228023047860327015, 8.226115314183148058787558262461
