| L(s) = 1 | + 9.16·3-s − 10.7·5-s + 7·7-s + 57.0·9-s − 11·11-s + 48.7·13-s − 98.2·15-s + 87.5·17-s − 35.4·19-s + 64.1·21-s + 215.·23-s − 10.1·25-s + 275.·27-s − 0.722·29-s + 270.·31-s − 100.·33-s − 75.0·35-s − 27.3·37-s + 447.·39-s − 381.·41-s − 137.·43-s − 611.·45-s − 202.·47-s + 49·49-s + 802.·51-s − 443.·53-s + 117.·55-s + ⋯ |
| L(s) = 1 | + 1.76·3-s − 0.958·5-s + 0.377·7-s + 2.11·9-s − 0.301·11-s + 1.04·13-s − 1.69·15-s + 1.24·17-s − 0.428·19-s + 0.666·21-s + 1.95·23-s − 0.0809·25-s + 1.96·27-s − 0.00462·29-s + 1.56·31-s − 0.532·33-s − 0.362·35-s − 0.121·37-s + 1.83·39-s − 1.45·41-s − 0.487·43-s − 2.02·45-s − 0.628·47-s + 0.142·49-s + 2.20·51-s − 1.14·53-s + 0.289·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.261586410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261586410\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 9.16T + 27T^{2} \) |
| 5 | \( 1 + 10.7T + 125T^{2} \) |
| 13 | \( 1 - 48.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 215.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.722T + 2.43e4T^{2} \) |
| 31 | \( 1 - 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 27.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 381.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 137.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 202.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 443.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 845.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 539.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 550.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 52.4T + 3.89e5T^{2} \) |
| 79 | \( 1 - 379.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 47.7T + 5.71e5T^{2} \) |
| 89 | \( 1 - 308.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 93.5T + 9.12e5T^{2} \) |
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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
−11.22933308702922281383284913452, −10.18412326019096677684076639157, −9.117083252204804305110323292738, −8.217702324866974658776318812029, −7.893132894470447775775084838663, −6.74022801205435713217253405932, −4.86432200548533070772898966671, −3.67724627246997365226688157246, −2.96899550574724924062191494725, −1.33881361221283851389659702196,
1.33881361221283851389659702196, 2.96899550574724924062191494725, 3.67724627246997365226688157246, 4.86432200548533070772898966671, 6.74022801205435713217253405932, 7.893132894470447775775084838663, 8.217702324866974658776318812029, 9.117083252204804305110323292738, 10.18412326019096677684076639157, 11.22933308702922281383284913452
