L(s) = 1 | + (2 + 2i)2-s + (2.64 − 2.64i)3-s + 8i·4-s + (22.4 + 11.0i)5-s + 10.5·6-s + (−62.5 − 62.5i)7-s + (−16 + 16i)8-s + 66.9i·9-s + (22.7 + 66.9i)10-s − 187.·11-s + (21.1 + 21.1i)12-s + (−63.2 + 63.2i)13-s − 250. i·14-s + (88.6 − 30.1i)15-s − 64·16-s + (−295. − 295. i)17-s + ⋯ |
L(s) = 1 | + (0.5 + 0.5i)2-s + (0.294 − 0.294i)3-s + 0.5i·4-s + (0.896 + 0.442i)5-s + 0.294·6-s + (−1.27 − 1.27i)7-s + (−0.250 + 0.250i)8-s + 0.826i·9-s + (0.227 + 0.669i)10-s − 1.55·11-s + (0.147 + 0.147i)12-s + (−0.374 + 0.374i)13-s − 1.27i·14-s + (0.394 − 0.133i)15-s − 0.250·16-s + (−1.02 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0953i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6444606478\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6444606478\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 - 2i)T \) |
| 5 | \( 1 + (-22.4 - 11.0i)T \) |
| 23 | \( 1 + (77.9 - 77.9i)T \) |
good | 3 | \( 1 + (-2.64 + 2.64i)T - 81iT^{2} \) |
| 7 | \( 1 + (62.5 + 62.5i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 187.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (63.2 - 63.2i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (295. + 295. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 514. iT - 1.30e5T^{2} \) |
| 29 | \( 1 - 672. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 333.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (724. + 724. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 556.T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-2.22e3 + 2.22e3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.75e3 + 1.75e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (2.26e3 - 2.26e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 2.96e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.24e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (5.38e3 + 5.38e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 4.78e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (2.03e3 - 2.03e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 4.85e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-4.03e3 + 4.03e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 7.29e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-4.62e3 - 4.62e3i)T + 8.85e7iT^{2} \) |
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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
−12.53247680374296321026060756595, −10.73733906223719042297744886299, −10.28818292873862727530245190835, −9.140643713786025048833068188464, −7.56999674848773106770763582698, −7.14215453779392779597008519299, −5.99186259062641521743491196680, −4.84899251769187442641525696507, −3.31700443135398800568014469503, −2.22287596700516759103443216061,
0.15418799809627862970207188228, 2.35999534940523581891432954399, 3.00019123415867539466798163316, 4.69974638124732152458742757887, 5.79342368148448028391674070633, 6.49789753373657016645018790248, 8.474068545103582426036980650408, 9.368683534506994450791726590835, 9.893806537827166434259479461868, 11.00021975620960119248453951713