L(s) = 1 | − 1.70·2-s + 3·3-s − 5.10·4-s − 5.10·6-s + 22.2·7-s + 22.2·8-s + 9·9-s − 1.79·11-s − 15.3·12-s + 58.2·13-s − 37.7·14-s + 2.89·16-s + 18.9·17-s − 15.3·18-s + 104.·19-s + 66.6·21-s + 3.04·22-s − 49.6·23-s + 66.8·24-s − 99.0·26-s + 27·27-s − 113.·28-s − 293.·29-s + 64.4·31-s − 183.·32-s − 5.37·33-s − 32.3·34-s + ⋯ |
L(s) = 1 | − 0.601·2-s + 0.577·3-s − 0.638·4-s − 0.347·6-s + 1.19·7-s + 0.985·8-s + 0.333·9-s − 0.0490·11-s − 0.368·12-s + 1.24·13-s − 0.721·14-s + 0.0452·16-s + 0.270·17-s − 0.200·18-s + 1.26·19-s + 0.692·21-s + 0.0295·22-s − 0.449·23-s + 0.568·24-s − 0.747·26-s + 0.192·27-s − 0.765·28-s − 1.87·29-s + 0.373·31-s − 1.01·32-s − 0.0283·33-s − 0.162·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.316930406\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.316930406\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 1.70T + 8T^{2} \) |
| 7 | \( 1 - 22.2T + 343T^{2} \) |
| 11 | \( 1 + 1.79T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 293.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 64.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 19.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 384.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 463.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 73.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 137.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 173.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 594.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 770.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 173.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 384.T + 9.12e5T^{2} \) |
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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
−13.96421613736799455626654224012, −13.30445648441860509218390676394, −11.67176241449830026330259873699, −10.52267456278578779951991535147, −9.300029839835695070702623422304, −8.358410719490616120529746773823, −7.53332712345290363728509280646, −5.35242682104027895694792299706, −3.86442451056307274839247791522, −1.42782798213752939473729709287,
1.42782798213752939473729709287, 3.86442451056307274839247791522, 5.35242682104027895694792299706, 7.53332712345290363728509280646, 8.358410719490616120529746773823, 9.300029839835695070702623422304, 10.52267456278578779951991535147, 11.67176241449830026330259873699, 13.30445648441860509218390676394, 13.96421613736799455626654224012