| L(s) = 1 | − i·2-s + i·3-s − 4-s + (−0.707 − 2.12i)5-s + 6-s + 0.414i·7-s + i·8-s − 9-s + (−2.12 + 0.707i)10-s − 3.58·11-s − i·12-s + 3.24i·13-s + 0.414·14-s + (2.12 − 0.707i)15-s + 16-s + 6.41i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (−0.316 − 0.948i)5-s + 0.408·6-s + 0.156i·7-s + 0.353i·8-s − 0.333·9-s + (−0.670 + 0.223i)10-s − 1.08·11-s − 0.288i·12-s + 0.899i·13-s + 0.110·14-s + (0.547 − 0.182i)15-s + 0.250·16-s + 1.55i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.192692676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.192692676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.707 + 2.12i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 - 0.414iT - 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 - 3.24iT - 13T^{2} \) |
| 17 | \( 1 - 6.41iT - 17T^{2} \) |
| 19 | \( 1 - 7.82T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 - 6.48iT - 43T^{2} \) |
| 47 | \( 1 - 7.89iT - 47T^{2} \) |
| 53 | \( 1 + 1.82iT - 53T^{2} \) |
| 59 | \( 1 + 1.07T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 4.82iT - 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 + 7.82iT - 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 18.7iT - 97T^{2} \) |
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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
−10.01079128490580501890539654583, −9.142380472235559301518603126922, −8.400096108453994437331774226043, −7.81547704204009221156921775500, −6.32793177543000704781684784022, −5.23158957079379639123550981183, −4.64455808985397001753157061976, −3.72706007252514306689074627160, −2.61178069832126933487519023709, −1.17396202243104779597542778735,
0.62232481732299538339438259789, 2.69575198636421418207435831836, 3.33398058283875100785931057948, 4.95383299185688125703484283755, 5.56494869353098323184789162774, 6.69640481852430767602328493362, 7.37662940428517991402144348949, 7.79228186136020868583679831790, 8.700854452398426219546243955134, 10.02739238190348016366115469451
