L(s) = 1 | − 9.40·2-s − 39.5·4-s + 482.·5-s + 1.33e3·7-s + 1.57e3·8-s − 4.53e3·10-s + 1.51e3·11-s − 1.42e4·13-s − 1.25e4·14-s − 9.75e3·16-s − 2.36e4·17-s − 4.11e4·19-s − 1.90e4·20-s − 1.42e4·22-s + 1.21e4·23-s + 1.54e5·25-s + 1.33e5·26-s − 5.26e4·28-s − 1.00e5·29-s − 2.49e5·31-s − 1.09e5·32-s + 2.22e5·34-s + 6.42e5·35-s + 1.84e5·37-s + 3.87e5·38-s + 7.60e5·40-s − 4.92e5·41-s + ⋯ |
L(s) = 1 | − 0.831·2-s − 0.309·4-s + 1.72·5-s + 1.46·7-s + 1.08·8-s − 1.43·10-s + 0.344·11-s − 1.79·13-s − 1.21·14-s − 0.595·16-s − 1.16·17-s − 1.37·19-s − 0.533·20-s − 0.286·22-s + 0.208·23-s + 1.98·25-s + 1.49·26-s − 0.453·28-s − 0.767·29-s − 1.50·31-s − 0.593·32-s + 0.971·34-s + 2.53·35-s + 0.598·37-s + 1.14·38-s + 1.87·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
good | 2 | \( 1 + 9.40T + 128T^{2} \) |
| 5 | \( 1 - 482.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.33e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.51e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.42e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.11e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.00e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.49e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.84e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.92e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.22e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.13e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.39e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.10e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.35e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.40e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.40e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 9.14e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.86e6T + 8.07e13T^{2} \) |
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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
−10.43739207963765723747773127995, −9.490972216113907693836934800749, −8.913496895968612311163678902830, −7.81654895617964514102710307414, −6.65593514910742772213459183954, −5.19770976512646251904071122316, −4.59069436495358774575336966281, −2.09981533344842857365066448294, −1.69898338941572940873563171625, 0,
1.69898338941572940873563171625, 2.09981533344842857365066448294, 4.59069436495358774575336966281, 5.19770976512646251904071122316, 6.65593514910742772213459183954, 7.81654895617964514102710307414, 8.913496895968612311163678902830, 9.490972216113907693836934800749, 10.43739207963765723747773127995