L(s) = 1 | + 2-s − i·3-s + 4-s + (2.18 − 0.452i)5-s − i·6-s − 0.0776i·7-s + 8-s − 9-s + (2.18 − 0.452i)10-s + 1.27·11-s − i·12-s + 1.98·13-s − 0.0776i·14-s + (−0.452 − 2.18i)15-s + 16-s − 2.25·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.979 − 0.202i)5-s − 0.408i·6-s − 0.0293i·7-s + 0.353·8-s − 0.333·9-s + (0.692 − 0.143i)10-s + 0.385·11-s − 0.288i·12-s + 0.550·13-s − 0.0207i·14-s + (−0.116 − 0.565i)15-s + 0.250·16-s − 0.545·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.076518182\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.076518182\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 5 | \( 1 + (-2.18 + 0.452i)T \) |
| 37 | \( 1 + (4.90 - 3.59i)T \) |
good | 7 | \( 1 + 0.0776iT - 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 - 1.98T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 2.64iT - 19T^{2} \) |
| 23 | \( 1 - 7.25T + 23T^{2} \) |
| 29 | \( 1 + 5.46iT - 29T^{2} \) |
| 31 | \( 1 + 6.55iT - 31T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 7.27T + 43T^{2} \) |
| 47 | \( 1 - 8.23iT - 47T^{2} \) |
| 53 | \( 1 - 4.77iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 - 9.19iT - 61T^{2} \) |
| 67 | \( 1 + 2.98iT - 67T^{2} \) |
| 71 | \( 1 - 7.16T + 71T^{2} \) |
| 73 | \( 1 - 9.35iT - 73T^{2} \) |
| 79 | \( 1 + 15.9iT - 79T^{2} \) |
| 83 | \( 1 + 3.25iT - 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.764617789279226332177739186906, −8.898827399672549928345431771083, −8.069499669808173456284949863773, −6.90689131716096446060404247507, −6.35623200547166125464641816126, −5.55873446882679281399576161991, −4.66150325984955267307175954300, −3.44348988734535192228661510853, −2.30718987049902350782229881055, −1.28899110641147382908189586986,
1.58379092683059452437102480819, 2.86787020169507607006872699168, 3.69496181014817586172509600838, 5.00513720486209389148583712496, 5.36340387405460003973922632761, 6.67425685872584799557506060219, 6.91886627146658898654952449443, 8.664406050473525763143910418500, 9.011011106635687865120471414158, 10.14628715543363898034526655562