L(s) = 1 | + 2-s + 2.61·3-s + 4-s + 0.316·5-s + 2.61·6-s − 4.23·7-s + 8-s + 3.85·9-s + 0.316·10-s + 2.97·11-s + 2.61·12-s + 3.99·13-s − 4.23·14-s + 0.827·15-s + 16-s + 4.57·17-s + 3.85·18-s + 1.78·19-s + 0.316·20-s − 11.0·21-s + 2.97·22-s + 23-s + 2.61·24-s − 4.90·25-s + 3.99·26-s + 2.22·27-s − 4.23·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.141·5-s + 1.06·6-s − 1.60·7-s + 0.353·8-s + 1.28·9-s + 0.0999·10-s + 0.897·11-s + 0.755·12-s + 1.10·13-s − 1.13·14-s + 0.213·15-s + 0.250·16-s + 1.10·17-s + 0.907·18-s + 0.409·19-s + 0.0706·20-s − 2.42·21-s + 0.634·22-s + 0.208·23-s + 0.534·24-s − 0.980·25-s + 0.784·26-s + 0.428·27-s − 0.800·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.147901594\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.147901594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 - 0.316T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 2.97T + 11T^{2} \) |
| 13 | \( 1 - 3.99T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 - 1.78T + 19T^{2} \) |
| 31 | \( 1 + 0.0125T + 31T^{2} \) |
| 37 | \( 1 + 8.58T + 37T^{2} \) |
| 41 | \( 1 - 5.02T + 41T^{2} \) |
| 43 | \( 1 + 5.02T + 43T^{2} \) |
| 47 | \( 1 + 4.22T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 8.71T + 61T^{2} \) |
| 67 | \( 1 - 9.45T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 3.99T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 + 8.37T + 89T^{2} \) |
| 97 | \( 1 - 5.75T + 97T^{2} \) |
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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
−9.649916319734872169090021732015, −8.849339782772852946745406891399, −8.062415485100264427760307326627, −7.07013449371422524308704224904, −6.38895093703613101648192053763, −5.53100700954809651117493626303, −3.91840633406307895763170959615, −3.52806972895713462626475977991, −2.83670437235886265413095341948, −1.50550447042976015404119915262,
1.50550447042976015404119915262, 2.83670437235886265413095341948, 3.52806972895713462626475977991, 3.91840633406307895763170959615, 5.53100700954809651117493626303, 6.38895093703613101648192053763, 7.07013449371422524308704224904, 8.062415485100264427760307326627, 8.849339782772852946745406891399, 9.649916319734872169090021732015