| L(s) = 1 | + 2.64·2-s + 5.00·4-s + 2.64·5-s − 2·7-s + 7.93·8-s + 7.00·10-s − 5·13-s − 5.29·14-s + 11.0·16-s + 2.64·17-s + 13.2·20-s + 5.29·23-s + 2.00·25-s − 13.2·26-s − 10.0·28-s − 7.93·29-s + 4·31-s + 13.2·32-s + 7.00·34-s − 5.29·35-s − 3·37-s + 21.0·40-s + 2.64·41-s − 6·43-s + 14.0·46-s − 10.5·47-s − 3·49-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.50·4-s + 1.18·5-s − 0.755·7-s + 2.80·8-s + 2.21·10-s − 1.38·13-s − 1.41·14-s + 2.75·16-s + 0.641·17-s + 2.95·20-s + 1.10·23-s + 0.400·25-s − 2.59·26-s − 1.88·28-s − 1.47·29-s + 0.718·31-s + 2.33·32-s + 1.20·34-s − 0.894·35-s − 0.493·37-s + 3.32·40-s + 0.413·41-s − 0.914·43-s + 2.06·46-s − 1.54·47-s − 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.514428828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.514428828\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.64T + 2T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 2.64T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 7.93T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 7.93T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−9.914965703692168903433585252577, −9.453377280534218477108632006757, −7.80500451405150098022293393114, −6.84591754041248826153327675164, −6.32708095088322834043551689914, −5.33759634431304167642847462094, −4.95387901592604198404956014156, −3.59897321520177948699551500908, −2.78790442035988167127374852458, −1.85912584640685147780352936602,
1.85912584640685147780352936602, 2.78790442035988167127374852458, 3.59897321520177948699551500908, 4.95387901592604198404956014156, 5.33759634431304167642847462094, 6.32708095088322834043551689914, 6.84591754041248826153327675164, 7.80500451405150098022293393114, 9.453377280534218477108632006757, 9.914965703692168903433585252577
