L(s) = 1 | + (1.94 − 0.447i)2-s − 3.19i·3-s + (3.59 − 1.74i)4-s − 3.90·5-s + (−1.43 − 6.23i)6-s + 2.64i·7-s + (6.23 − 5.01i)8-s − 1.23·9-s + (−7.60 + 1.74i)10-s + 13.2i·11-s + (−5.58 − 11.5i)12-s + 1.60·13-s + (1.18 + 5.15i)14-s + 12.4i·15-s + (9.91 − 12.5i)16-s + 8.10·17-s + ⋯ |
L(s) = 1 | + (0.974 − 0.223i)2-s − 1.06i·3-s + (0.899 − 0.436i)4-s − 0.780·5-s + (−0.238 − 1.03i)6-s + 0.378i·7-s + (0.779 − 0.626i)8-s − 0.137·9-s + (−0.760 + 0.174i)10-s + 1.20i·11-s + (−0.465 − 0.959i)12-s + 0.123·13-s + (0.0845 + 0.368i)14-s + 0.832i·15-s + (0.619 − 0.784i)16-s + 0.476·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.436 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.67127 - 1.04718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67127 - 1.04718i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.94 + 0.447i)T \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 + 3.19iT - 9T^{2} \) |
| 5 | \( 1 + 3.90T + 25T^{2} \) |
| 7 | \( 1 - 2.64iT - 49T^{2} \) |
| 11 | \( 1 - 13.2iT - 121T^{2} \) |
| 13 | \( 1 - 1.60T + 169T^{2} \) |
| 17 | \( 1 - 8.10T + 289T^{2} \) |
| 23 | \( 1 - 38.2iT - 529T^{2} \) |
| 29 | \( 1 + 51.1T + 841T^{2} \) |
| 31 | \( 1 + 13.6iT - 961T^{2} \) |
| 37 | \( 1 + 35.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.06T + 1.68e3T^{2} \) |
| 43 | \( 1 + 16.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 1.59iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 56.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 106. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 35.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 47.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 34.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 71.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 149. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 169.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 109.T + 9.40e3T^{2} \) |
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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
−13.81487907523145300141753307341, −12.85783837680268847019466082082, −12.13445015173975691464552008637, −11.34532093716463591474445593196, −9.736925690919939146036487800368, −7.69718132623534560604355200338, −7.06172075844459048601310830793, −5.51882181003969479708021739931, −3.86245796768298382754820290981, −1.89647913782104993580287036651,
3.43291263748503298481147151087, 4.31427087735724631054194832323, 5.73038517966097620047823738092, 7.33525850884920181571913644323, 8.631660007393844863658705836710, 10.39840963082615081694052426746, 11.15758244361419508997785488936, 12.29578474942570283597827426532, 13.55144010006253019445719085047, 14.61817478156520369964272926220