| L(s) = 1 | + 2.72·2-s − 3-s + 5.40·4-s + 1.58·5-s − 2.72·6-s − 7-s + 9.26·8-s + 9-s + 4.30·10-s + 5.54·11-s − 5.40·12-s − 2.72·14-s − 1.58·15-s + 14.4·16-s + 4.80·17-s + 2.72·18-s − 0.892·19-s + 8.55·20-s + 21-s + 15.0·22-s − 4.63·23-s − 9.26·24-s − 2.49·25-s − 27-s − 5.40·28-s − 1.88·29-s − 4.30·30-s + ⋯ |
| L(s) = 1 | + 1.92·2-s − 0.577·3-s + 2.70·4-s + 0.707·5-s − 1.11·6-s − 0.377·7-s + 3.27·8-s + 0.333·9-s + 1.36·10-s + 1.67·11-s − 1.56·12-s − 0.727·14-s − 0.408·15-s + 3.60·16-s + 1.16·17-s + 0.641·18-s − 0.204·19-s + 1.91·20-s + 0.218·21-s + 3.21·22-s − 0.965·23-s − 1.89·24-s − 0.498·25-s − 0.192·27-s − 1.02·28-s − 0.349·29-s − 0.786·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.967749413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.967749413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.72T + 2T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 17 | \( 1 - 4.80T + 17T^{2} \) |
| 19 | \( 1 + 0.892T + 19T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 + 1.88T + 29T^{2} \) |
| 31 | \( 1 + 1.47T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 + 9.75T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 6.25T + 53T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 61 | \( 1 + 3.58T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 5.33T + 73T^{2} \) |
| 79 | \( 1 - 0.779T + 79T^{2} \) |
| 83 | \( 1 + 1.47T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 0.758T + 97T^{2} \) |
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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
−8.373224070293037148432400320750, −7.29692624725560519740024507862, −6.59905493180006417569419895620, −6.19303915245347742457042549853, −5.52113659333560086714944197627, −4.88849046193096937845982068719, −3.75048500135486079711501540869, −3.57475101792177818890114530789, −2.14942692196400078186336772792, −1.41805335174726634507012207150,
1.41805335174726634507012207150, 2.14942692196400078186336772792, 3.57475101792177818890114530789, 3.75048500135486079711501540869, 4.88849046193096937845982068719, 5.52113659333560086714944197627, 6.19303915245347742457042549853, 6.59905493180006417569419895620, 7.29692624725560519740024507862, 8.373224070293037148432400320750
