L(s) = 1 | + 2.46·2-s + 3.13·3-s + 4.09·4-s + 5-s + 7.73·6-s + 0.465·7-s + 5.17·8-s + 6.80·9-s + 2.46·10-s − 11-s + 12.8·12-s − 3.06·13-s + 1.15·14-s + 3.13·15-s + 4.58·16-s − 5.55·17-s + 16.8·18-s − 1.70·19-s + 4.09·20-s + 1.45·21-s − 2.46·22-s + 8.11·23-s + 16.2·24-s + 25-s − 7.57·26-s + 11.9·27-s + 1.90·28-s + ⋯ |
L(s) = 1 | + 1.74·2-s + 1.80·3-s + 2.04·4-s + 0.447·5-s + 3.15·6-s + 0.176·7-s + 1.83·8-s + 2.26·9-s + 0.780·10-s − 0.301·11-s + 3.70·12-s − 0.850·13-s + 0.307·14-s + 0.808·15-s + 1.14·16-s − 1.34·17-s + 3.96·18-s − 0.391·19-s + 0.916·20-s + 0.318·21-s − 0.526·22-s + 1.69·23-s + 3.30·24-s + 0.200·25-s − 1.48·26-s + 2.29·27-s + 0.360·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.06257747\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.06257747\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 3 | \( 1 - 3.13T + 3T^{2} \) |
| 7 | \( 1 - 0.465T + 7T^{2} \) |
| 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 + 5.55T + 17T^{2} \) |
| 19 | \( 1 + 1.70T + 19T^{2} \) |
| 23 | \( 1 - 8.11T + 23T^{2} \) |
| 29 | \( 1 + 4.76T + 29T^{2} \) |
| 31 | \( 1 - 2.39T + 31T^{2} \) |
| 37 | \( 1 + 8.51T + 37T^{2} \) |
| 41 | \( 1 - 9.76T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 5.47T + 47T^{2} \) |
| 53 | \( 1 - 6.18T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 3.02T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 6.17T + 71T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 + 6.58T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 9.66T + 97T^{2} \) |
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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
−8.465512582898940197807437151613, −7.38008429110763475826828668482, −7.06092738795348681578073929035, −6.21575738243098558699896917329, −5.03117942302453566340504829650, −4.65788876077552103298052803118, −3.76127870049083593533908346362, −3.01263679644555608636451719913, −2.39255847429338467271025723658, −1.78364579382949066669623882982,
1.78364579382949066669623882982, 2.39255847429338467271025723658, 3.01263679644555608636451719913, 3.76127870049083593533908346362, 4.65788876077552103298052803118, 5.03117942302453566340504829650, 6.21575738243098558699896917329, 7.06092738795348681578073929035, 7.38008429110763475826828668482, 8.465512582898940197807437151613