L(s) = 1 | + 1.77i·5-s + (−2.39 − 1.12i)7-s + 3.85·11-s + 6.59·13-s − 4.56·17-s + 1.05·19-s + 0.0946i·23-s + 1.83·25-s − 4.31·29-s − 4.07i·31-s + (2.00 − 4.25i)35-s + 4.65i·37-s + 11.5·41-s − 6.28i·43-s − 6.12·47-s + ⋯ |
L(s) = 1 | + 0.794i·5-s + (−0.904 − 0.426i)7-s + 1.16·11-s + 1.82·13-s − 1.10·17-s + 0.241·19-s + 0.0197i·23-s + 0.367·25-s − 0.801·29-s − 0.732i·31-s + (0.339 − 0.718i)35-s + 0.764i·37-s + 1.80·41-s − 0.958i·43-s − 0.894·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.926924867\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.926924867\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.39 + 1.12i)T \) |
good | 5 | \( 1 - 1.77iT - 5T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 - 6.59T + 13T^{2} \) |
| 17 | \( 1 + 4.56T + 17T^{2} \) |
| 19 | \( 1 - 1.05T + 19T^{2} \) |
| 23 | \( 1 - 0.0946iT - 23T^{2} \) |
| 29 | \( 1 + 4.31T + 29T^{2} \) |
| 31 | \( 1 + 4.07iT - 31T^{2} \) |
| 37 | \( 1 - 4.65iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 6.28iT - 43T^{2} \) |
| 47 | \( 1 + 6.12T + 47T^{2} \) |
| 53 | \( 1 + 2.44T + 53T^{2} \) |
| 59 | \( 1 + 13.1iT - 59T^{2} \) |
| 61 | \( 1 + 4.58T + 61T^{2} \) |
| 67 | \( 1 + 4.83iT - 67T^{2} \) |
| 71 | \( 1 - 11.5iT - 71T^{2} \) |
| 73 | \( 1 + 2.25iT - 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 - 10.6iT - 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 + 14.2iT - 97T^{2} \) |
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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
−8.588904931810939196911301985114, −7.63545221723533191742230039220, −6.73765649304049860492152397846, −6.44829738299366080721888660853, −5.83075068573620301419500914022, −4.46239950756022766623165344275, −3.70570525353924992467813788356, −3.24580377577196518027465602636, −2.02297603698654514082493916375, −0.819279796801785433657231064657,
0.823800134712241787522434729941, 1.75860430053663209926993988971, 3.05126636985044112035694481485, 3.85566362376786111461043981119, 4.49993199578868682517708202126, 5.60029604518944487617462136141, 6.25547399997666918908697304746, 6.69840875553257351177103052895, 7.76890128847686931481505857012, 8.730071147383543877491463450025