L(s) = 1 | − 6.89·2-s + 25.8·3-s + 15.5·4-s − 25·5-s − 178.·6-s + 221.·7-s + 113.·8-s + 423.·9-s + 172.·10-s + 121·11-s + 402.·12-s − 1.05e3·13-s − 1.52e3·14-s − 645.·15-s − 1.27e3·16-s − 1.49e3·17-s − 2.92e3·18-s + 361·19-s − 389.·20-s + 5.70e3·21-s − 834.·22-s + 4.34e3·23-s + 2.92e3·24-s + 625·25-s + 7.29e3·26-s + 4.66e3·27-s + 3.44e3·28-s + ⋯ |
L(s) = 1 | − 1.21·2-s + 1.65·3-s + 0.486·4-s − 0.447·5-s − 2.01·6-s + 1.70·7-s + 0.625·8-s + 1.74·9-s + 0.545·10-s + 0.301·11-s + 0.806·12-s − 1.73·13-s − 2.07·14-s − 0.740·15-s − 1.24·16-s − 1.25·17-s − 2.12·18-s + 0.229·19-s − 0.217·20-s + 2.82·21-s − 0.367·22-s + 1.71·23-s + 1.03·24-s + 0.200·25-s + 2.11·26-s + 1.23·27-s + 0.830·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 6.89T + 32T^{2} \) |
| 3 | \( 1 - 25.8T + 243T^{2} \) |
| 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.05e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.49e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.87e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.82e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.81e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.10e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.81e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.60e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.70e3T + 8.58e9T^{2} \) |
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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
−8.989884777984473501315615030448, −7.967972587643721026665764170202, −7.61132021924865515518258016585, −7.01533118574172348844696636422, −4.76613901591415485378838959650, −4.55674168220592260750821378023, −3.07234426362954902509230472938, −2.02126162364509271147605070410, −1.46217146989338402308000417481, 0,
1.46217146989338402308000417481, 2.02126162364509271147605070410, 3.07234426362954902509230472938, 4.55674168220592260750821378023, 4.76613901591415485378838959650, 7.01533118574172348844696636422, 7.61132021924865515518258016585, 7.967972587643721026665764170202, 8.989884777984473501315615030448