L(s) = 1 | + 17.8·2-s + 192.·4-s − 255.·5-s − 1.07e3·7-s + 1.15e3·8-s − 4.57e3·10-s − 4.74e3·11-s + 5.72e3·13-s − 1.92e4·14-s − 3.99e3·16-s + 1.28e4·17-s − 5.21e4·19-s − 4.91e4·20-s − 8.48e4·22-s + 9.82e3·23-s − 1.29e4·25-s + 1.02e5·26-s − 2.06e5·28-s + 1.96e5·29-s + 2.72e5·31-s − 2.19e5·32-s + 2.30e5·34-s + 2.73e5·35-s + 3.09e5·37-s − 9.32e5·38-s − 2.94e5·40-s + 4.21e5·41-s + ⋯ |
L(s) = 1 | + 1.58·2-s + 1.50·4-s − 0.913·5-s − 1.18·7-s + 0.796·8-s − 1.44·10-s − 1.07·11-s + 0.722·13-s − 1.87·14-s − 0.243·16-s + 0.634·17-s − 1.74·19-s − 1.37·20-s − 1.69·22-s + 0.168·23-s − 0.165·25-s + 1.14·26-s − 1.77·28-s + 1.49·29-s + 1.64·31-s − 1.18·32-s + 1.00·34-s + 1.08·35-s + 1.00·37-s − 2.75·38-s − 0.727·40-s + 0.954·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(2.942700806\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.942700806\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
good | 2 | \( 1 - 17.8T + 128T^{2} \) |
| 5 | \( 1 + 255.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.07e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.74e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.72e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.28e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 5.21e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 9.82e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.96e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.72e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.09e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.21e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.61e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.38e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.33e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.03e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.88e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.34e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.12e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.32e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.46e7T + 8.07e13T^{2} \) |
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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
−9.997635993228845144294837008739, −8.602068908460455139915952687442, −7.74643439625549229723709433230, −6.49775850126206883264624196917, −6.11013617373486887483683007649, −4.83256692298377482963196025428, −4.07017995241208080367417322300, −3.20538350208371886063168761766, −2.48321009316638548293329689749, −0.55254259129790807462441121762,
0.55254259129790807462441121762, 2.48321009316638548293329689749, 3.20538350208371886063168761766, 4.07017995241208080367417322300, 4.83256692298377482963196025428, 6.11013617373486887483683007649, 6.49775850126206883264624196917, 7.74643439625549229723709433230, 8.602068908460455139915952687442, 9.997635993228845144294837008739