| L(s) = 1 | + 2i·2-s − 4·4-s + 32i·7-s − 8i·8-s + 60·11-s + 34i·13-s − 64·14-s + 16·16-s − 42i·17-s + 76·19-s + 120i·22-s − 68·26-s − 128i·28-s + 6·29-s − 232·31-s + 32i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.72i·7-s − 0.353i·8-s + 1.64·11-s + 0.725i·13-s − 1.22·14-s + 0.250·16-s − 0.599i·17-s + 0.917·19-s + 1.16i·22-s − 0.512·26-s − 0.863i·28-s + 0.0384·29-s − 1.34·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.700499078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700499078\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 32iT - 343T^{2} \) |
| 11 | \( 1 - 60T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 42iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 76T + 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 232T + 2.97e4T^{2} \) |
| 37 | \( 1 - 134iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 234T + 6.89e4T^{2} \) |
| 43 | \( 1 - 412iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 360iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 222iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 660T + 2.05e5T^{2} \) |
| 61 | \( 1 + 490T + 2.26e5T^{2} \) |
| 67 | \( 1 - 812iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 120T + 3.57e5T^{2} \) |
| 73 | \( 1 + 746iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 152T + 4.93e5T^{2} \) |
| 83 | \( 1 + 804iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 678T + 7.04e5T^{2} \) |
| 97 | \( 1 - 194iT - 9.12e5T^{2} \) |
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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
−11.41634391908890801855662568750, −9.628846829947841016113485089352, −9.200758631150008014799649441554, −8.504650111796543161488609042482, −7.24285252541707020506546643841, −6.34031533826960366872582671706, −5.56132103344646327153951873948, −4.47828883224516753490115609939, −3.10864917349973933654939251243, −1.56047244359457693387529961113,
0.59461831291699184534200897269, 1.61124761801604400487030980137, 3.57013973419933974095619786190, 3.94683224588907942385580563854, 5.31839359331918009536537271833, 6.71176049185541299580077309044, 7.48408825553197541344651142689, 8.629838067292762559393991131701, 9.635198483667549174820465745977, 10.36960486207413218562429654310
