L(s) = 1 | + 2-s + 4-s − 3.46i·5-s − i·7-s + 8-s − 3.46i·10-s + (−0.748 − 3.23i)11-s + 2.27i·13-s − i·14-s + 16-s − 1.31·17-s − 1.04i·19-s − 3.46i·20-s + (−0.748 − 3.23i)22-s − 2.56i·23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.54i·5-s − 0.377i·7-s + 0.353·8-s − 1.09i·10-s + (−0.225 − 0.974i)11-s + 0.630i·13-s − 0.267i·14-s + 0.250·16-s − 0.319·17-s − 0.239i·19-s − 0.774i·20-s + (−0.159 − 0.688i)22-s − 0.534i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.250488104\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.250488104\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.748 + 3.23i)T \) |
good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 13 | \( 1 - 2.27iT - 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 + 1.04iT - 19T^{2} \) |
| 23 | \( 1 + 2.56iT - 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 - 2.87T + 31T^{2} \) |
| 37 | \( 1 + 5.87T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 - 4.69iT - 43T^{2} \) |
| 47 | \( 1 + 12.8iT - 47T^{2} \) |
| 53 | \( 1 - 3.57iT - 53T^{2} \) |
| 59 | \( 1 - 4.94iT - 59T^{2} \) |
| 61 | \( 1 + 9.19iT - 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 - 4.36iT - 71T^{2} \) |
| 73 | \( 1 + 14.2iT - 73T^{2} \) |
| 79 | \( 1 - 16.8iT - 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 6.49iT - 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.115176413324595649703122121389, −8.596629000662571214961230520108, −7.76433688826348242856998706895, −6.70802709027005965901584425843, −5.82858845267489174611461685046, −4.98318984952568146195906234739, −4.36480241617334788401523586392, −3.40460160975681412022000310152, −1.98180194817367139699403886542, −0.70006251466714807309504223689,
1.99729119947617918379566184533, 2.84725552392348183327291162514, 3.65998085566804491564921110698, 4.77851494541237571166939218771, 5.76764698450626696563389456213, 6.49392055733300504185725017113, 7.30513880094200375346576496708, 7.83310010909455378488815780622, 9.142534648796565260894041997477, 10.14043137754234207671136816324