L(s) = 1 | + (−0.227 − 0.227i)5-s − 3.90·7-s + (−2.21 + 2.21i)11-s + (−5.08 + 5.08i)13-s − 18.8·17-s + (11.7 + 11.7i)19-s + 35.4·23-s − 24.8i·25-s + (21.2 − 21.2i)29-s − 35.9i·31-s + (0.888 + 0.888i)35-s + (34.4 + 34.4i)37-s − 44.1i·41-s + (−28.3 + 28.3i)43-s − 32.8i·47-s + ⋯ |
L(s) = 1 | + (−0.0455 − 0.0455i)5-s − 0.557·7-s + (−0.200 + 0.200i)11-s + (−0.391 + 0.391i)13-s − 1.10·17-s + (0.619 + 0.619i)19-s + 1.54·23-s − 0.995i·25-s + (0.732 − 0.732i)29-s − 1.16i·31-s + (0.0253 + 0.0253i)35-s + (0.930 + 0.930i)37-s − 1.07i·41-s + (−0.658 + 0.658i)43-s − 0.699i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.332896052\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332896052\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.227 + 0.227i)T + 25iT^{2} \) |
| 7 | \( 1 + 3.90T + 49T^{2} \) |
| 11 | \( 1 + (2.21 - 2.21i)T - 121iT^{2} \) |
| 13 | \( 1 + (5.08 - 5.08i)T - 169iT^{2} \) |
| 17 | \( 1 + 18.8T + 289T^{2} \) |
| 19 | \( 1 + (-11.7 - 11.7i)T + 361iT^{2} \) |
| 23 | \( 1 - 35.4T + 529T^{2} \) |
| 29 | \( 1 + (-21.2 + 21.2i)T - 841iT^{2} \) |
| 31 | \( 1 + 35.9iT - 961T^{2} \) |
| 37 | \( 1 + (-34.4 - 34.4i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 44.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (28.3 - 28.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 32.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (42.1 + 42.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-66.9 + 66.9i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (17.3 - 17.3i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-39.6 - 39.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 63.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 75.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 59.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-71.2 - 71.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 150. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 51.5T + 9.40e3T^{2} \) |
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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
−9.615156371968246893252852023943, −8.649636566036146485508503276899, −7.85131986023291873422858018211, −6.84016749098568745502404181338, −6.27819196354858514576966575824, −5.07305668451475812817151496559, −4.29156399460191837394139385008, −3.11787974267501333547457614019, −2.10355931156884716347124253333, −0.47500696600616039366038209664,
1.01752902233240513997141686689, 2.64629414115574507326123554966, 3.36150429363936709847795543814, 4.70972089549762370351070282714, 5.38245785000414982534300933945, 6.61630052149826606146068730414, 7.09774543078606980372713951666, 8.138960276866631169728950982129, 9.099507646756756993498042003991, 9.553945085982666905244199687582