L(s) = 1 | + 1.54·2-s + 0.373·4-s + 3.37·7-s − 2.50·8-s + 0.275·11-s + 5.74·13-s + 5.20·14-s − 4.60·16-s + 6.11·17-s − 5.04·19-s + 0.424·22-s + 2.66·23-s + 8.84·26-s + 1.26·28-s − 6.27·29-s + 0.987·31-s − 2.08·32-s + 9.42·34-s + 6.20·37-s − 7.77·38-s − 0.876·41-s + 0.0269·43-s + 0.103·44-s + 4.10·46-s + 11.2·47-s + 4.39·49-s + 2.14·52-s + ⋯ |
L(s) = 1 | + 1.08·2-s + 0.186·4-s + 1.27·7-s − 0.885·8-s + 0.0830·11-s + 1.59·13-s + 1.39·14-s − 1.15·16-s + 1.48·17-s − 1.15·19-s + 0.0904·22-s + 0.555·23-s + 1.73·26-s + 0.238·28-s − 1.16·29-s + 0.177·31-s − 0.369·32-s + 1.61·34-s + 1.02·37-s − 1.26·38-s − 0.136·41-s + 0.00411·43-s + 0.0155·44-s + 0.605·46-s + 1.64·47-s + 0.628·49-s + 0.297·52-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
\(L(1)\) |
\(\approx\) |
\(4.077163248\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.077163248\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.54T + 2T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 - 0.275T + 11T^{2} \) |
| 13 | \( 1 - 5.74T + 13T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 + 5.04T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 6.27T + 29T^{2} \) |
| 31 | \( 1 - 0.987T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 + 0.876T + 41T^{2} \) |
| 43 | \( 1 - 0.0269T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 2.16T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 5.33T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 + 5.03T + 71T^{2} \) |
| 73 | \( 1 + 4.45T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 8.71T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 5.72T + 97T^{2} \) |
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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
−8.101432904273222978745554271557, −7.47201948619205689140238965134, −6.34951145948544354905654444770, −5.84994433258908546309474608715, −5.21472806794436230309854302102, −4.43089681826411813268323361071, −3.85392526862285963836566898241, −3.11141477014327572870662606186, −1.96763787789768698915611364939, −0.974745797067286622191734586043,
0.974745797067286622191734586043, 1.96763787789768698915611364939, 3.11141477014327572870662606186, 3.85392526862285963836566898241, 4.43089681826411813268323361071, 5.21472806794436230309854302102, 5.84994433258908546309474608715, 6.34951145948544354905654444770, 7.47201948619205689140238965134, 8.101432904273222978745554271557