| L(s) = 1 | + i·2-s − 4-s + (2.12 + 1.58i)7-s − i·8-s − 1.41i·11-s − 3.16i·13-s + (−1.58 + 2.12i)14-s + 16-s − 4.47·17-s + 1.41·22-s − 6i·23-s + 3.16·26-s + (−2.12 − 1.58i)28-s − 2.82i·29-s + i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.801 + 0.597i)7-s − 0.353i·8-s − 0.426i·11-s − 0.877i·13-s + (−0.422 + 0.566i)14-s + 0.250·16-s − 1.08·17-s + 0.301·22-s − 1.25i·23-s + 0.620·26-s + (−0.400 − 0.298i)28-s − 0.525i·29-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9093473441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9093473441\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.12 - 1.58i)T \) |
| good | 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 + 3.16iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 4.24T + 37T^{2} \) |
| 41 | \( 1 + 9.48T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 + 13.4iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 - 6.32iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 - 12.6iT - 97T^{2} \) |
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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
−8.227275144805085392486336367891, −8.163848677187609331124557371588, −6.86188085313094879766607364070, −6.40827197693951956320761867116, −5.35423191107027993362325851204, −4.97222725885207290654678758740, −3.97328138668293698745903473401, −2.88894076939788356151535083180, −1.81882865060693184619663308319, −0.27390117608502847315366615123,
1.41810944225660202738530486471, 2.02416664421675885721315756439, 3.28973003205975117040239598282, 4.16941980832217075306604336860, 4.76304671383827737858832066118, 5.56077836872511160154322366381, 6.84402298729645556629080897001, 7.21926009475423263146366344584, 8.333343723581300875638228525634, 8.790326492929052579661291597052
