L(s) = 1 | − 2·2-s + 7.28·3-s + 4·4-s − 14.5·6-s − 8·8-s + 26.0·9-s + 41.1·11-s + 29.1·12-s + 24.5·13-s + 16·16-s + 93.3·17-s − 52.0·18-s + 54.6·19-s − 82.2·22-s + 136.·23-s − 58.2·24-s − 49.1·26-s − 7.11·27-s + 282.·29-s − 54.6·31-s − 32·32-s + 299.·33-s − 186.·34-s + 104.·36-s − 212.·37-s − 109.·38-s + 179.·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.990·6-s − 0.353·8-s + 0.963·9-s + 1.12·11-s + 0.700·12-s + 0.524·13-s + 0.250·16-s + 1.33·17-s − 0.681·18-s + 0.659·19-s − 0.796·22-s + 1.23·23-s − 0.495·24-s − 0.370·26-s − 0.0507·27-s + 1.81·29-s − 0.316·31-s − 0.176·32-s + 1.57·33-s − 0.942·34-s + 0.481·36-s − 0.943·37-s − 0.466·38-s + 0.735·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.686668929\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.686668929\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 7.28T + 27T^{2} \) |
| 11 | \( 1 - 41.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 93.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 136.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 282.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 193.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 126.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 419.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 57.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 687.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 716.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 236.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 457.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.83e3T + 9.12e5T^{2} \) |
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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
−8.678983416511753066570393841287, −8.011365211511479327541923580322, −7.31045600247057977011444573836, −6.58024822523047338828379985053, −5.59041875458284939187060121300, −4.36869180332694983051765130769, −3.30250858723989247657846670325, −2.95910438581192222113092554154, −1.62626585717505399475814276201, −0.973276918638944429877813337597,
0.973276918638944429877813337597, 1.62626585717505399475814276201, 2.95910438581192222113092554154, 3.30250858723989247657846670325, 4.36869180332694983051765130769, 5.59041875458284939187060121300, 6.58024822523047338828379985053, 7.31045600247057977011444573836, 8.011365211511479327541923580322, 8.678983416511753066570393841287