L(s) = 1 | − 1.41i·2-s − 0.894i·3-s − 2.00·4-s + (−3.14 + 3.88i)5-s − 1.26·6-s − 4.24·7-s + 2.82i·8-s + 8.19·9-s + (5.49 + 4.44i)10-s + 9.15i·11-s + 1.78i·12-s + 6.01i·13-s + 6.00i·14-s + (3.47 + 2.81i)15-s + 4.00·16-s + 19.3·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.298i·3-s − 0.500·4-s + (−0.628 + 0.777i)5-s − 0.210·6-s − 0.606·7-s + 0.353i·8-s + 0.911·9-s + (0.549 + 0.444i)10-s + 0.832i·11-s + 0.149i·12-s + 0.462i·13-s + 0.428i·14-s + (0.231 + 0.187i)15-s + 0.250·16-s + 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01680 + 0.351627i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01680 + 0.351627i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + (3.14 - 3.88i)T \) |
| 23 | \( 1 + (5.12 - 22.4i)T \) |
good | 3 | \( 1 + 0.894iT - 9T^{2} \) |
| 7 | \( 1 + 4.24T + 49T^{2} \) |
| 11 | \( 1 - 9.15iT - 121T^{2} \) |
| 13 | \( 1 - 6.01iT - 169T^{2} \) |
| 17 | \( 1 - 19.3T + 289T^{2} \) |
| 19 | \( 1 - 31.0iT - 361T^{2} \) |
| 29 | \( 1 + 45.5T + 841T^{2} \) |
| 31 | \( 1 - 33.3T + 961T^{2} \) |
| 37 | \( 1 + 21.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 33.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.50T + 1.84e3T^{2} \) |
| 47 | \( 1 - 71.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 3.73T + 2.80e3T^{2} \) |
| 59 | \( 1 + 4.90T + 3.48e3T^{2} \) |
| 61 | \( 1 + 42.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 118.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 92.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 43.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 58.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 18.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 164. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 91.9T + 9.40e3T^{2} \) |
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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
−12.23703461052362519831257332713, −11.19211349088565211167369620582, −10.02328705138338306500656577598, −9.668762784659177119134144676276, −7.907417050086197035508700763125, −7.25708617800935097759031422602, −5.98558104257132967030959594843, −4.25086153240409782266633547238, −3.33404797015793350100095475971, −1.68649665527226901116464499504,
0.62074663065994631652362148290, 3.40573944686072169516121465117, 4.54859535704675350063919066767, 5.60224188999116564372326176831, 6.90462531993795187022434978599, 7.88269159707013443273633321562, 8.868648412512319567309125617892, 9.711257930064469939192510891950, 10.81222473644235750401460139368, 12.08108365622826543509557071382