| L(s) = 1 | − 3-s + 1.16·5-s + 7-s + 9-s − 6.56·11-s − 1.77·13-s − 1.16·15-s + 4.94·17-s − 1.64·19-s − 21-s − 23-s − 3.64·25-s − 27-s − 0.671·29-s − 2.94·31-s + 6.56·33-s + 1.16·35-s − 0.671·37-s + 1.77·39-s + 3.91·41-s + 10.3·43-s + 1.16·45-s − 8.81·47-s + 49-s − 4.94·51-s + 5.05·53-s − 7.64·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.520·5-s + 0.377·7-s + 0.333·9-s − 1.97·11-s − 0.493·13-s − 0.300·15-s + 1.19·17-s − 0.377·19-s − 0.218·21-s − 0.208·23-s − 0.728·25-s − 0.192·27-s − 0.124·29-s − 0.528·31-s + 1.14·33-s + 0.196·35-s − 0.110·37-s + 0.284·39-s + 0.611·41-s + 1.57·43-s + 0.173·45-s − 1.28·47-s + 0.142·49-s − 0.692·51-s + 0.694·53-s − 1.03·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.301595537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301595537\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 1.16T + 5T^{2} \) |
| 11 | \( 1 + 6.56T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 + 1.64T + 19T^{2} \) |
| 29 | \( 1 + 0.671T + 29T^{2} \) |
| 31 | \( 1 + 2.94T + 31T^{2} \) |
| 37 | \( 1 + 0.671T + 37T^{2} \) |
| 41 | \( 1 - 3.91T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 8.81T + 47T^{2} \) |
| 53 | \( 1 - 5.05T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 6.98T + 67T^{2} \) |
| 71 | \( 1 - 9.62T + 71T^{2} \) |
| 73 | \( 1 - 9.47T + 73T^{2} \) |
| 79 | \( 1 - 9.48T + 79T^{2} \) |
| 83 | \( 1 - 0.614T + 83T^{2} \) |
| 89 | \( 1 + 1.08T + 89T^{2} \) |
| 97 | \( 1 - 0.463T + 97T^{2} \) |
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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
−7.63600096713859772368976002528, −7.46464050248327204287230613800, −6.26212390464719376970623684520, −5.67367978127690572224310577505, −5.20178010440599914575947186589, −4.56009555060597325119929796750, −3.49318939230459470393832313341, −2.53357074697566990631712485802, −1.86453261746021599497412808038, −0.56794689726355308826718346030,
0.56794689726355308826718346030, 1.86453261746021599497412808038, 2.53357074697566990631712485802, 3.49318939230459470393832313341, 4.56009555060597325119929796750, 5.20178010440599914575947186589, 5.67367978127690572224310577505, 6.26212390464719376970623684520, 7.46464050248327204287230613800, 7.63600096713859772368976002528
