L(s) = 1 | + 0.533·2-s + 3-s − 1.71·4-s − 1.09·5-s + 0.533·6-s − 1.98·8-s + 9-s − 0.583·10-s − 3.05·11-s − 1.71·12-s − 4.92·13-s − 1.09·15-s + 2.37·16-s + 0.748·17-s + 0.533·18-s + 1.24·19-s + 1.87·20-s − 1.63·22-s − 23-s − 1.98·24-s − 3.80·25-s − 2.62·26-s + 27-s + 4.90·29-s − 0.583·30-s + 8.55·31-s + 5.23·32-s + ⋯ |
L(s) = 1 | + 0.377·2-s + 0.577·3-s − 0.857·4-s − 0.489·5-s + 0.217·6-s − 0.701·8-s + 0.333·9-s − 0.184·10-s − 0.921·11-s − 0.495·12-s − 1.36·13-s − 0.282·15-s + 0.592·16-s + 0.181·17-s + 0.125·18-s + 0.284·19-s + 0.419·20-s − 0.347·22-s − 0.208·23-s − 0.404·24-s − 0.760·25-s − 0.515·26-s + 0.192·27-s + 0.910·29-s − 0.106·30-s + 1.53·31-s + 0.924·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.457431368\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457431368\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 0.533T + 2T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 4.92T + 13T^{2} \) |
| 17 | \( 1 - 0.748T + 17T^{2} \) |
| 19 | \( 1 - 1.24T + 19T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 - 8.55T + 31T^{2} \) |
| 37 | \( 1 - 5.89T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 43 | \( 1 + 2.45T + 43T^{2} \) |
| 47 | \( 1 - 5.39T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 4.12T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 - 3.21T + 67T^{2} \) |
| 71 | \( 1 + 5.13T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 2.24T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.67T + 97T^{2} \) |
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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
−8.476906499184357520425632858720, −7.929882340939475097301420513487, −7.38581638829660892190599362673, −6.30547278499939884074486067466, −5.31281206821198389478453191899, −4.71659651130715055173905489280, −4.02707109178584770451008582728, −3.06399701808452138095063257058, −2.37394240642366791808495220936, −0.64154992193545169012728136860,
0.64154992193545169012728136860, 2.37394240642366791808495220936, 3.06399701808452138095063257058, 4.02707109178584770451008582728, 4.71659651130715055173905489280, 5.31281206821198389478453191899, 6.30547278499939884074486067466, 7.38581638829660892190599362673, 7.929882340939475097301420513487, 8.476906499184357520425632858720