| L(s) = 1 | + 3-s − 0.149·5-s − 1.16·7-s + 9-s + 11-s + 13-s − 0.149·15-s − 3.82·17-s + 1.16·19-s − 1.16·21-s − 3.67·23-s − 4.97·25-s + 27-s + 5.31·29-s − 0.187·31-s + 33-s + 0.174·35-s + 6.84·37-s + 39-s + 8.19·41-s + 0.806·43-s − 0.149·45-s − 6.80·47-s − 5.63·49-s − 3.82·51-s + 8.63·53-s − 0.149·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.0667·5-s − 0.441·7-s + 0.333·9-s + 0.301·11-s + 0.277·13-s − 0.0385·15-s − 0.928·17-s + 0.268·19-s − 0.254·21-s − 0.767·23-s − 0.995·25-s + 0.192·27-s + 0.987·29-s − 0.0336·31-s + 0.174·33-s + 0.0294·35-s + 1.12·37-s + 0.160·39-s + 1.27·41-s + 0.123·43-s − 0.0222·45-s − 0.993·47-s − 0.805·49-s − 0.536·51-s + 1.18·53-s − 0.0201·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.250589041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.250589041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 0.149T + 5T^{2} \) |
| 7 | \( 1 + 1.16T + 7T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 - 1.16T + 19T^{2} \) |
| 23 | \( 1 + 3.67T + 23T^{2} \) |
| 29 | \( 1 - 5.31T + 29T^{2} \) |
| 31 | \( 1 + 0.187T + 31T^{2} \) |
| 37 | \( 1 - 6.84T + 37T^{2} \) |
| 41 | \( 1 - 8.19T + 41T^{2} \) |
| 43 | \( 1 - 0.806T + 43T^{2} \) |
| 47 | \( 1 + 6.80T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 4.84T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 5.53T + 79T^{2} \) |
| 83 | \( 1 - 5.32T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.027299708564292780260356472608, −7.34384377680970021521546442856, −6.49130675500590981556297009540, −6.08402307485031824726287828585, −5.03966982951047791708801915383, −4.16832162339560426192469399898, −3.66778144986140093851163587239, −2.68047653950582472381750918225, −1.98264727290800365870137635631, −0.73745117382435819634120873554,
0.73745117382435819634120873554, 1.98264727290800365870137635631, 2.68047653950582472381750918225, 3.66778144986140093851163587239, 4.16832162339560426192469399898, 5.03966982951047791708801915383, 6.08402307485031824726287828585, 6.49130675500590981556297009540, 7.34384377680970021521546442856, 8.027299708564292780260356472608
