L(s) = 1 | + (1 − i)2-s + (2.20 + 2.20i)3-s − 2i·4-s + (0.183 + 4.99i)5-s + 4.40·6-s + (−8.35 + 8.35i)7-s + (−2 − 2i)8-s + 0.680i·9-s + (5.18 + 4.81i)10-s + 4.85·11-s + (4.40 − 4.40i)12-s + (17.1 + 17.1i)13-s + 16.7i·14-s + (−10.5 + 11.3i)15-s − 4·16-s + (−9.16 + 9.16i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (0.733 + 0.733i)3-s − 0.5i·4-s + (0.0367 + 0.999i)5-s + 0.733·6-s + (−1.19 + 1.19i)7-s + (−0.250 − 0.250i)8-s + 0.0756i·9-s + (0.518 + 0.481i)10-s + 0.441·11-s + (0.366 − 0.366i)12-s + (1.32 + 1.32i)13-s + 1.19i·14-s + (−0.705 + 0.759i)15-s − 0.250·16-s + (−0.538 + 0.538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.96777 + 1.14497i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96777 + 1.14497i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 5 | \( 1 + (-0.183 - 4.99i)T \) |
| 23 | \( 1 + (-3.39 - 3.39i)T \) |
good | 3 | \( 1 + (-2.20 - 2.20i)T + 9iT^{2} \) |
| 7 | \( 1 + (8.35 - 8.35i)T - 49iT^{2} \) |
| 11 | \( 1 - 4.85T + 121T^{2} \) |
| 13 | \( 1 + (-17.1 - 17.1i)T + 169iT^{2} \) |
| 17 | \( 1 + (9.16 - 9.16i)T - 289iT^{2} \) |
| 19 | \( 1 + 28.3iT - 361T^{2} \) |
| 29 | \( 1 + 16.7iT - 841T^{2} \) |
| 31 | \( 1 - 49.9T + 961T^{2} \) |
| 37 | \( 1 + (15.0 - 15.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 43.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (5.23 + 5.23i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-20.2 + 20.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (34.0 + 34.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 67.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 10.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-18.0 + 18.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 21.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (21.4 + 21.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 113. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (45.6 + 45.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 64.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (98.3 - 98.3i)T - 9.40e3iT^{2} \) |
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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
−11.96806402833933879935176065320, −11.27689025096083318359372119629, −10.14809095223063939007612875223, −9.234246328391852102243392957963, −8.779593853406271906867306797362, −6.59690932511793735054891481680, −6.20524733406188394335387037869, −4.28228849999623214369950720333, −3.32104218938399515332167885847, −2.43660738001541820037376869231,
1.05283907216392655886422230615, 3.16556355810644575196519521859, 4.19912528916000362895756497280, 5.77559162411590781018396701332, 6.77759354681984208821062731478, 7.86150685595382595220277012388, 8.501866937904742695389223203367, 9.699432730373960441758814604351, 10.85183622378365838293942436045, 12.52180459557469572207922053182