L(s) = 1 | − 0.732·3-s + 5-s − 3.46·7-s − 2.46·9-s + 1.26·11-s − 13-s − 0.732·15-s − 3.46·17-s + 6.73·19-s + 2.53·21-s + 2.19·23-s + 25-s + 4·27-s − 1.46·29-s − 1.26·31-s − 0.928·33-s − 3.46·35-s − 10.9·37-s + 0.732·39-s − 4.53·41-s − 0.732·43-s − 2.46·45-s − 4.53·47-s + 4.99·49-s + 2.53·51-s − 4.53·53-s + 1.26·55-s + ⋯ |
L(s) = 1 | − 0.422·3-s + 0.447·5-s − 1.30·7-s − 0.821·9-s + 0.382·11-s − 0.277·13-s − 0.189·15-s − 0.840·17-s + 1.54·19-s + 0.553·21-s + 0.457·23-s + 0.200·25-s + 0.769·27-s − 0.271·29-s − 0.227·31-s − 0.161·33-s − 0.585·35-s − 1.79·37-s + 0.117·39-s − 0.708·41-s − 0.111·43-s − 0.367·45-s − 0.661·47-s + 0.714·49-s + 0.355·51-s − 0.623·53-s + 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.055018178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.055018178\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 + 0.732T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 + 4.53T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 1.46T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.66T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 4.53T + 83T^{2} \) |
| 89 | \( 1 - 8.92T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−8.684098497123236333134664111833, −7.50474927349745960477494028340, −6.74120531314950521785448352380, −6.32143856864762125515502736658, −5.43772059586355040663616277739, −4.93688029176900716759460633098, −3.55915821608018156637579337505, −3.09918837941087504382649412719, −2.00021103727726203443023403309, −0.57363547950088717604790874780,
0.57363547950088717604790874780, 2.00021103727726203443023403309, 3.09918837941087504382649412719, 3.55915821608018156637579337505, 4.93688029176900716759460633098, 5.43772059586355040663616277739, 6.32143856864762125515502736658, 6.74120531314950521785448352380, 7.50474927349745960477494028340, 8.684098497123236333134664111833