L(s) = 1 | + 1.32·2-s + 4.35·3-s − 6.25·4-s + 15.6·5-s + 5.75·6-s − 4.97·7-s − 18.8·8-s − 8.03·9-s + 20.6·10-s + 3.73·11-s − 27.2·12-s + 34.7·13-s − 6.57·14-s + 67.9·15-s + 25.1·16-s − 90.1·17-s − 10.6·18-s − 19.2·19-s − 97.6·20-s − 21.6·21-s + 4.93·22-s + 34.2·23-s − 81.9·24-s + 118.·25-s + 45.8·26-s − 152.·27-s + 31.1·28-s + ⋯ |
L(s) = 1 | + 0.466·2-s + 0.838·3-s − 0.782·4-s + 1.39·5-s + 0.391·6-s − 0.268·7-s − 0.831·8-s − 0.297·9-s + 0.651·10-s + 0.102·11-s − 0.655·12-s + 0.740·13-s − 0.125·14-s + 1.17·15-s + 0.393·16-s − 1.28·17-s − 0.138·18-s − 0.232·19-s − 1.09·20-s − 0.225·21-s + 0.0477·22-s + 0.310·23-s − 0.697·24-s + 0.949·25-s + 0.345·26-s − 1.08·27-s + 0.210·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 1.32T + 8T^{2} \) |
| 3 | \( 1 - 4.35T + 27T^{2} \) |
| 5 | \( 1 - 15.6T + 125T^{2} \) |
| 7 | \( 1 + 4.97T + 343T^{2} \) |
| 11 | \( 1 - 3.73T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 90.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 19.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 34.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 29.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 224.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 487.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 114.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 683.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 370.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 854.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 475.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 879.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 670.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 115.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 939.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 598.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 664.T + 9.12e5T^{2} \) |
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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
−8.862740319572363627084492902462, −7.926763931736680852390500580386, −6.63676399384123353125944198386, −5.98990416833556814461628315889, −5.29929784340219691863743954897, −4.29148401448280683264932735251, −3.38096201130160459686092328705, −2.54632492235886654249484286575, −1.57846501861201597423048413932, 0,
1.57846501861201597423048413932, 2.54632492235886654249484286575, 3.38096201130160459686092328705, 4.29148401448280683264932735251, 5.29929784340219691863743954897, 5.98990416833556814461628315889, 6.63676399384123353125944198386, 7.926763931736680852390500580386, 8.862740319572363627084492902462