| L(s) = 1 | − 2-s + 4-s + 4.12·7-s − 8-s + 2.64·11-s + 2.51·13-s − 4.12·14-s + 16-s + 0.515·17-s + 19-s − 2.64·22-s − 3.09·23-s − 2.51·26-s + 4.12·28-s + 7.79·29-s + 3.67·31-s − 32-s − 0.515·34-s + 10.2·37-s − 38-s − 8.88·41-s + 8.64·43-s + 2.64·44-s + 3.09·46-s + 4.96·47-s + 10.0·49-s + 2.51·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.55·7-s − 0.353·8-s + 0.795·11-s + 0.697·13-s − 1.10·14-s + 0.250·16-s + 0.124·17-s + 0.229·19-s − 0.562·22-s − 0.645·23-s − 0.493·26-s + 0.779·28-s + 1.44·29-s + 0.659·31-s − 0.176·32-s − 0.0883·34-s + 1.68·37-s − 0.162·38-s − 1.38·41-s + 1.31·43-s + 0.397·44-s + 0.456·46-s + 0.724·47-s + 1.43·49-s + 0.348·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.255709165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.255709165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4.12T + 7T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 - 0.515T + 17T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 8.88T + 41T^{2} \) |
| 43 | \( 1 - 8.64T + 43T^{2} \) |
| 47 | \( 1 - 4.96T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 + 3.15T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 2.70T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 7.60T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−7.74313073441995241768519881422, −7.45266803653456056503766104389, −6.23770355235695113068603570636, −6.06986418251652417888396084053, −4.82775545867929385239795881132, −4.42029328367395343590608225728, −3.41375804390819842166373868506, −2.41466119881694596788898804192, −1.50974072729124478760353690694, −0.921017425114854591707469467823,
0.921017425114854591707469467823, 1.50974072729124478760353690694, 2.41466119881694596788898804192, 3.41375804390819842166373868506, 4.42029328367395343590608225728, 4.82775545867929385239795881132, 6.06986418251652417888396084053, 6.23770355235695113068603570636, 7.45266803653456056503766104389, 7.74313073441995241768519881422
