L(s) = 1 | + (−10.1 − 17.5i)5-s + (15.4 + 8.92i)11-s − 33.1i·13-s + (−22.9 + 39.6i)17-s + (35.0 − 20.2i)19-s + (69.7 − 40.2i)23-s + (−142. + 246. i)25-s + 233. i·29-s + (195. + 112. i)31-s + (135. + 234. i)37-s − 154.·41-s + 367.·43-s + (−263. − 457. i)47-s + (−78.8 − 45.5i)53-s − 361. i·55-s + ⋯ |
L(s) = 1 | + (−0.905 − 1.56i)5-s + (0.423 + 0.244i)11-s − 0.707i·13-s + (−0.326 + 0.566i)17-s + (0.422 − 0.244i)19-s + (0.632 − 0.365i)23-s + (−1.14 + 1.97i)25-s + 1.49i·29-s + (1.13 + 0.653i)31-s + (0.601 + 1.04i)37-s − 0.588·41-s + 1.30·43-s + (−0.818 − 1.41i)47-s + (−0.204 − 0.118i)53-s − 0.885i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.671339541\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.671339541\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (10.1 + 17.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-15.4 - 8.92i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 33.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (22.9 - 39.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-35.0 + 20.2i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-69.7 + 40.2i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 233. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-195. - 112. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-135. - 234. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (263. + 457. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (78.8 + 45.5i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (312. - 541. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-78.8 + 45.5i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (431. - 747. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 303. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-999. - 576. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-3.48 - 6.02i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 815.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-155. - 269. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.83e3iT - 9.12e5T^{2} \) |
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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
−8.679002479271549280842268132943, −8.318045597726308575012671307153, −7.42776777606173374273564436885, −6.54748846350029614748758887013, −5.34128201836434086398227186265, −4.80170969188942572804978127834, −4.00239115603665675501426684158, −3.02364402600970471363193161967, −1.43468066596024235559428212192, −0.70679213864043055150697903442,
0.56930978941114828214716432564, 2.21248842945257438069406270117, 3.07711619903289293507538239368, 3.88420759967191496570872716496, 4.68580036189144487632285470410, 6.17846758048863695602243830129, 6.52026990450515356604189550921, 7.59501700783363494493504687833, 7.81528847957977816716722116686, 9.122495926865593817806908089690