| L(s) = 1 | − 0.326·2-s − 1.89·4-s − 3.42·7-s + 1.27·8-s + 5.34·11-s + 3.52·13-s + 1.11·14-s + 3.37·16-s + 2.55·17-s − 2.02·19-s − 1.74·22-s + 7.57·23-s − 1.15·26-s + 6.48·28-s − 4.74·29-s + 1.62·31-s − 3.64·32-s − 0.835·34-s + 0.0134·37-s + 0.661·38-s − 9.67·41-s − 2.32·43-s − 10.1·44-s − 2.47·46-s + 6.94·47-s + 4.72·49-s − 6.66·52-s + ⋯ |
| L(s) = 1 | − 0.231·2-s − 0.946·4-s − 1.29·7-s + 0.449·8-s + 1.61·11-s + 0.976·13-s + 0.299·14-s + 0.842·16-s + 0.620·17-s − 0.464·19-s − 0.372·22-s + 1.57·23-s − 0.225·26-s + 1.22·28-s − 0.880·29-s + 0.291·31-s − 0.644·32-s − 0.143·34-s + 0.00220·37-s + 0.107·38-s − 1.51·41-s − 0.354·43-s − 1.52·44-s − 0.364·46-s + 1.01·47-s + 0.674·49-s − 0.924·52-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
| \(L(1)\) |
\(\approx\) |
\(1.300345894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.300345894\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.326T + 2T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 3.52T + 13T^{2} \) |
| 17 | \( 1 - 2.55T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 - 7.57T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 - 1.62T + 31T^{2} \) |
| 37 | \( 1 - 0.0134T + 37T^{2} \) |
| 41 | \( 1 + 9.67T + 41T^{2} \) |
| 43 | \( 1 + 2.32T + 43T^{2} \) |
| 47 | \( 1 - 6.94T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 - 0.0221T + 59T^{2} \) |
| 61 | \( 1 - 3.91T + 61T^{2} \) |
| 67 | \( 1 + 4.11T + 67T^{2} \) |
| 71 | \( 1 + 2.33T + 71T^{2} \) |
| 73 | \( 1 + 1.51T + 73T^{2} \) |
| 79 | \( 1 - 0.426T + 79T^{2} \) |
| 83 | \( 1 + 6.04T + 83T^{2} \) |
| 89 | \( 1 - 6.09T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.437355858378982978657217919843, −7.34534238306995828127130501408, −6.68990723594396741605148492597, −6.08270292618528036205715348966, −5.30276528535194722733948708496, −4.30064145647072326897340965055, −3.63679422784959757609325929455, −3.16180359781300870300279453539, −1.55300250571897353200863921113, −0.68414859331185977490223710408,
0.68414859331185977490223710408, 1.55300250571897353200863921113, 3.16180359781300870300279453539, 3.63679422784959757609325929455, 4.30064145647072326897340965055, 5.30276528535194722733948708496, 6.08270292618528036205715348966, 6.68990723594396741605148492597, 7.34534238306995828127130501408, 8.437355858378982978657217919843
