| L(s) = 1 | − 2·2-s + 0.589·3-s + 4·4-s − 5·5-s − 1.17·6-s − 18.5·7-s − 8·8-s − 26.6·9-s + 10·10-s + 47.9·11-s + 2.35·12-s + 42.3·13-s + 37.0·14-s − 2.94·15-s + 16·16-s + 1.70·17-s + 53.3·18-s + 21.4·19-s − 20·20-s − 10.9·21-s − 95.8·22-s − 23·23-s − 4.71·24-s + 25·25-s − 84.7·26-s − 31.6·27-s − 74.0·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.113·3-s + 0.5·4-s − 0.447·5-s − 0.0802·6-s − 0.999·7-s − 0.353·8-s − 0.987·9-s + 0.316·10-s + 1.31·11-s + 0.0567·12-s + 0.903·13-s + 0.706·14-s − 0.0507·15-s + 0.250·16-s + 0.0243·17-s + 0.697·18-s + 0.258·19-s − 0.223·20-s − 0.113·21-s − 0.928·22-s − 0.208·23-s − 0.0401·24-s + 0.200·25-s − 0.639·26-s − 0.225·27-s − 0.499·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.047376531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047376531\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 - 0.589T + 27T^{2} \) |
| 7 | \( 1 + 18.5T + 343T^{2} \) |
| 11 | \( 1 - 47.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 42.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.70T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.4T + 6.85e3T^{2} \) |
| 29 | \( 1 - 57.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 7.85T + 5.06e4T^{2} \) |
| 41 | \( 1 - 465.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 182.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 449.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 368.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 377.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 849.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 92.3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 626.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 439.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 641.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.12e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.42e3T + 9.12e5T^{2} \) |
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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
−11.65836451076832049138511530701, −10.81505583792425751126455375858, −9.588277649748678898892067227271, −8.901443017497042459704749871343, −7.987328015995729355076230620203, −6.66228505681297785097783241915, −5.96938972579766822693923211827, −3.97657288497232002418100685636, −2.85032150788752427345480824765, −0.841714331942519847881041423366,
0.841714331942519847881041423366, 2.85032150788752427345480824765, 3.97657288497232002418100685636, 5.96938972579766822693923211827, 6.66228505681297785097783241915, 7.987328015995729355076230620203, 8.901443017497042459704749871343, 9.588277649748678898892067227271, 10.81505583792425751126455375858, 11.65836451076832049138511530701
