L(s) = 1 | + 1.31i·3-s − 4.66i·7-s + 1.28·9-s + 2.23·11-s + 2.80i·13-s + 7.63i·17-s + 1.36·19-s + 6.11·21-s + i·23-s + 5.61i·27-s − 8.94·29-s − 1.58·31-s + 2.92i·33-s − 1.40i·37-s − 3.67·39-s + ⋯ |
L(s) = 1 | + 0.756i·3-s − 1.76i·7-s + 0.427·9-s + 0.672·11-s + 0.776i·13-s + 1.85i·17-s + 0.312·19-s + 1.33·21-s + 0.208i·23-s + 1.08i·27-s − 1.66·29-s − 0.284·31-s + 0.508i·33-s − 0.230i·37-s − 0.587·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.003244383\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.003244383\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 - 1.31iT - 3T^{2} \) |
| 7 | \( 1 + 4.66iT - 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 - 2.80iT - 13T^{2} \) |
| 17 | \( 1 - 7.63iT - 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 29 | \( 1 + 8.94T + 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + 1.40iT - 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 7.26iT - 47T^{2} \) |
| 53 | \( 1 - 8.38iT - 53T^{2} \) |
| 59 | \( 1 - 4.88T + 59T^{2} \) |
| 61 | \( 1 - 4.33T + 61T^{2} \) |
| 67 | \( 1 - 8.54iT - 67T^{2} \) |
| 71 | \( 1 - 8.81T + 71T^{2} \) |
| 73 | \( 1 - 5.26iT - 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 4.59iT - 83T^{2} \) |
| 89 | \( 1 - 4.70T + 89T^{2} \) |
| 97 | \( 1 - 16.5iT - 97T^{2} \) |
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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
−8.505061765054588851108319743664, −7.52742504459699548411118370462, −7.13676288720742529447345006168, −6.34256224339705982486113808174, −5.45222655530906820712655691444, −4.33289829200181995809376117745, −3.96446488149724385083121161740, −3.61244621265953966085035186770, −1.89470826237627409253672751881, −1.06603649084009464945521913997,
0.63553877813517176601706157036, 1.86562988398739534014016613601, 2.56131912838282152944323167143, 3.39898344429363603666580101598, 4.62877795525431083702393581842, 5.41329439019315019849777051697, 5.97509477264130439324293005347, 6.77473781280979679131802754463, 7.50856975412289937179382661329, 8.044029595932870537669937530330