L(s) = 1 | + 1.53·2-s + 0.369·4-s − 3.87·7-s − 2.51·8-s − 1.24·11-s + 13-s − 5.97·14-s − 4.60·16-s + 0.659·17-s + 6.97·19-s − 1.92·22-s + 1.55·23-s + 1.53·26-s − 1.43·28-s + 3·29-s + 5.43·31-s − 2.06·32-s + 1.01·34-s − 2.29·37-s + 10.7·38-s + 10.2·41-s + 8.20·43-s − 0.460·44-s + 2.38·46-s + 1.53·47-s + 8.04·49-s + 0.369·52-s + ⋯ |
L(s) = 1 | + 1.08·2-s + 0.184·4-s − 1.46·7-s − 0.887·8-s − 0.376·11-s + 0.277·13-s − 1.59·14-s − 1.15·16-s + 0.160·17-s + 1.59·19-s − 0.409·22-s + 0.323·23-s + 0.301·26-s − 0.270·28-s + 0.557·29-s + 0.975·31-s − 0.364·32-s + 0.174·34-s − 0.376·37-s + 1.74·38-s + 1.59·41-s + 1.25·43-s − 0.0694·44-s + 0.352·46-s + 0.224·47-s + 1.14·49-s + 0.0511·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.209248038\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.209248038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 + 1.24T + 11T^{2} \) |
| 17 | \( 1 - 0.659T + 17T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 - 1.55T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 5.43T + 31T^{2} \) |
| 37 | \( 1 + 2.29T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 8.20T + 43T^{2} \) |
| 47 | \( 1 - 1.53T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.20T + 73T^{2} \) |
| 79 | \( 1 + 0.474T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−8.995862090684997692520030256467, −7.84975827217032439810029636920, −7.03540010666930334630083928021, −6.19721620610135034730443725838, −5.72467087680512764869443512393, −4.85928834100505225150399426191, −3.97869927936541648161338984453, −3.15718491759773245631852631260, −2.69217201980668938075989564551, −0.76127228644676281525429094865,
0.76127228644676281525429094865, 2.69217201980668938075989564551, 3.15718491759773245631852631260, 3.97869927936541648161338984453, 4.85928834100505225150399426191, 5.72467087680512764869443512393, 6.19721620610135034730443725838, 7.03540010666930334630083928021, 7.84975827217032439810029636920, 8.995862090684997692520030256467