| L(s) = 1 | − 0.0581·2-s − 3-s − 1.99·4-s − 0.0629·5-s + 0.0581·6-s − 7-s + 0.232·8-s + 9-s + 0.00366·10-s + 0.305·11-s + 1.99·12-s + 13-s + 0.0581·14-s + 0.0629·15-s + 3.97·16-s − 17-s − 0.0581·18-s − 7.95·19-s + 0.125·20-s + 21-s − 0.0177·22-s + 8.88·23-s − 0.232·24-s − 4.99·25-s − 0.0581·26-s − 27-s + 1.99·28-s + ⋯ |
| L(s) = 1 | − 0.0411·2-s − 0.577·3-s − 0.998·4-s − 0.0281·5-s + 0.0237·6-s − 0.377·7-s + 0.0821·8-s + 0.333·9-s + 0.00115·10-s + 0.0920·11-s + 0.576·12-s + 0.277·13-s + 0.0155·14-s + 0.0162·15-s + 0.994·16-s − 0.242·17-s − 0.0137·18-s − 1.82·19-s + 0.0281·20-s + 0.218·21-s − 0.00378·22-s + 1.85·23-s − 0.0474·24-s − 0.999·25-s − 0.0114·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4641 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7193911990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7193911990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0581T + 2T^{2} \) |
| 5 | \( 1 + 0.0629T + 5T^{2} \) |
| 11 | \( 1 - 0.305T + 11T^{2} \) |
| 19 | \( 1 + 7.95T + 19T^{2} \) |
| 23 | \( 1 - 8.88T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 + 9.02T + 31T^{2} \) |
| 37 | \( 1 - 6.91T + 37T^{2} \) |
| 41 | \( 1 + 5.60T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 + 1.82T + 47T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 + 1.09T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 - 6.54T + 67T^{2} \) |
| 71 | \( 1 + 9.78T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 0.440T + 79T^{2} \) |
| 83 | \( 1 - 3.65T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 0.504T + 97T^{2} \) |
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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
−8.522673105403034559839851373377, −7.56117982343266402483853441206, −6.80741183962868278557752165137, −6.03733323940474062885715504868, −5.39889291363444835970847020014, −4.51092294737736419273657548131, −4.01600145445895524699793455471, −3.03722615108608572288574367826, −1.73040427350705375275127795728, −0.49415804255880779890921703391,
0.49415804255880779890921703391, 1.73040427350705375275127795728, 3.03722615108608572288574367826, 4.01600145445895524699793455471, 4.51092294737736419273657548131, 5.39889291363444835970847020014, 6.03733323940474062885715504868, 6.80741183962868278557752165137, 7.56117982343266402483853441206, 8.522673105403034559839851373377
