| L(s) = 1 | − 0.899·2-s − 1.19·4-s − 3.85·7-s + 2.86·8-s + 4.88·11-s + 1.48·13-s + 3.47·14-s − 0.198·16-s + 3.74·17-s + 7.15·19-s − 4.39·22-s + 6.63·23-s − 1.33·26-s + 4.59·28-s + 8.70·29-s − 1.54·31-s − 5.56·32-s − 3.36·34-s − 2.52·37-s − 6.43·38-s − 0.951·41-s − 6.07·43-s − 5.81·44-s − 5.96·46-s − 3.79·47-s + 7.89·49-s − 1.76·52-s + ⋯ |
| L(s) = 1 | − 0.635·2-s − 0.595·4-s − 1.45·7-s + 1.01·8-s + 1.47·11-s + 0.411·13-s + 0.927·14-s − 0.0497·16-s + 0.908·17-s + 1.64·19-s − 0.936·22-s + 1.38·23-s − 0.261·26-s + 0.868·28-s + 1.61·29-s − 0.277·31-s − 0.983·32-s − 0.577·34-s − 0.415·37-s − 1.04·38-s − 0.148·41-s − 0.926·43-s − 0.877·44-s − 0.879·46-s − 0.554·47-s + 1.12·49-s − 0.245·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.248056946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248056946\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.899T + 2T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 - 3.74T + 17T^{2} \) |
| 19 | \( 1 - 7.15T + 19T^{2} \) |
| 23 | \( 1 - 6.63T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 + 0.951T + 41T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 + 3.79T + 47T^{2} \) |
| 53 | \( 1 - 7.19T + 53T^{2} \) |
| 59 | \( 1 + 4.20T + 59T^{2} \) |
| 61 | \( 1 + 5.70T + 61T^{2} \) |
| 67 | \( 1 - 0.532T + 67T^{2} \) |
| 71 | \( 1 + 0.452T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 - 4.98T + 79T^{2} \) |
| 83 | \( 1 + 7.29T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 2.65T + 97T^{2} \) |
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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
−8.387628736102924525736881333052, −7.32462637772364473548496563748, −6.89153311955054255900902846451, −6.11124166422399386503304973669, −5.28452075810717410734832539749, −4.43255022383600932921649628374, −3.40459073295948854299700999133, −3.17694423000139082896707008480, −1.39272758154985931418746878702, −0.75786843073452784267184614122,
0.75786843073452784267184614122, 1.39272758154985931418746878702, 3.17694423000139082896707008480, 3.40459073295948854299700999133, 4.43255022383600932921649628374, 5.28452075810717410734832539749, 6.11124166422399386503304973669, 6.89153311955054255900902846451, 7.32462637772364473548496563748, 8.387628736102924525736881333052
