L(s) = 1 | + 16.3·2-s − 27·3-s + 137.·4-s + 361.·5-s − 440.·6-s + 685.·7-s + 158.·8-s + 729·9-s + 5.88e3·10-s − 2.07e3·11-s − 3.71e3·12-s − 1.38e3·13-s + 1.11e4·14-s − 9.75e3·15-s − 1.50e4·16-s − 1.49e4·17-s + 1.18e4·18-s + 5.09e4·19-s + 4.97e4·20-s − 1.85e4·21-s − 3.38e4·22-s + 1.64e4·23-s − 4.27e3·24-s + 5.24e4·25-s − 2.25e4·26-s − 1.96e4·27-s + 9.44e4·28-s + ⋯ |
L(s) = 1 | + 1.44·2-s − 0.577·3-s + 1.07·4-s + 1.29·5-s − 0.831·6-s + 0.755·7-s + 0.109·8-s + 0.333·9-s + 1.86·10-s − 0.469·11-s − 0.621·12-s − 0.174·13-s + 1.08·14-s − 0.746·15-s − 0.918·16-s − 0.736·17-s + 0.480·18-s + 1.70·19-s + 1.39·20-s − 0.436·21-s − 0.677·22-s + 0.282·23-s − 0.0631·24-s + 0.670·25-s − 0.251·26-s − 0.192·27-s + 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(6.140254821\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.140254821\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 27T \) |
| 157 | \( 1 + 3.86e6T \) |
good | 2 | \( 1 - 16.3T + 128T^{2} \) |
| 5 | \( 1 - 361.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 685.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.07e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.49e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.09e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.64e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.04e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.80e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.62e3T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.99e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.52e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.52e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.39e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.92e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.93e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.88e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.59e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.97e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.92e6T + 8.07e13T^{2} \) |
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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
−10.02979437716324320671453419950, −9.172353105181265695271876711796, −7.79483913205666795742177329888, −6.56971882599783591024809737704, −5.93094010871939748460415452127, −5.01954431605973569764679504408, −4.63614110953644549119115044295, −3.06054182814397123409630401979, −2.15959239260740852059470834778, −0.942750034520433726819742337424,
0.942750034520433726819742337424, 2.15959239260740852059470834778, 3.06054182814397123409630401979, 4.63614110953644549119115044295, 5.01954431605973569764679504408, 5.93094010871939748460415452127, 6.56971882599783591024809737704, 7.79483913205666795742177329888, 9.172353105181265695271876711796, 10.02979437716324320671453419950