L(s) = 1 | + (1.26 − 1.26i)3-s + (2.95 + 2.95i)5-s − i·7-s − 0.189i·9-s + (3.18 + 3.18i)11-s + (3.42 − 3.42i)13-s + 7.46·15-s − 5.13·17-s + (1.50 − 1.50i)19-s + (−1.26 − 1.26i)21-s − 7.11i·23-s + 12.4i·25-s + (3.54 + 3.54i)27-s + (−3.84 + 3.84i)29-s + 0.831·31-s + ⋯ |
L(s) = 1 | + (0.729 − 0.729i)3-s + (1.32 + 1.32i)5-s − 0.377i·7-s − 0.0630i·9-s + (0.960 + 0.960i)11-s + (0.950 − 0.950i)13-s + 1.92·15-s − 1.24·17-s + (0.345 − 0.345i)19-s + (−0.275 − 0.275i)21-s − 1.48i·23-s + 2.49i·25-s + (0.683 + 0.683i)27-s + (−0.714 + 0.714i)29-s + 0.149·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.074295897\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.074295897\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-1.26 + 1.26i)T - 3iT^{2} \) |
| 5 | \( 1 + (-2.95 - 2.95i)T + 5iT^{2} \) |
| 11 | \( 1 + (-3.18 - 3.18i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.42 + 3.42i)T - 13iT^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 + (-1.50 + 1.50i)T - 19iT^{2} \) |
| 23 | \( 1 + 7.11iT - 23T^{2} \) |
| 29 | \( 1 + (3.84 - 3.84i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.831T + 31T^{2} \) |
| 37 | \( 1 + (5.64 + 5.64i)T + 37iT^{2} \) |
| 41 | \( 1 - 2.22iT - 41T^{2} \) |
| 43 | \( 1 + (1.61 + 1.61i)T + 43iT^{2} \) |
| 47 | \( 1 - 7.83T + 47T^{2} \) |
| 53 | \( 1 + (-5.58 - 5.58i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.85 + 1.85i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.65 - 1.65i)T - 61iT^{2} \) |
| 67 | \( 1 + (5.77 - 5.77i)T - 67iT^{2} \) |
| 71 | \( 1 - 6.04iT - 71T^{2} \) |
| 73 | \( 1 - 7.67iT - 73T^{2} \) |
| 79 | \( 1 + 1.90T + 79T^{2} \) |
| 83 | \( 1 + (-7.97 + 7.97i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.49iT - 89T^{2} \) |
| 97 | \( 1 + 1.98T + 97T^{2} \) |
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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
−9.128946727549909186939783170922, −8.677125325990571059678302893561, −7.41507883677721193357947737135, −6.96009725075671880236505670062, −6.39640236161823202122561666542, −5.45891214510020246346927944487, −4.14627813322403923869530086592, −3.01052682070373641773708955386, −2.28960239255906098971651544694, −1.47118416131107862470698144663,
1.23294813043114005955042735017, 2.11144665083620146468923570586, 3.51724071445458897687831182703, 4.19243443609202263792846521148, 5.17751266378942565255487568570, 6.02872946461038674262110563710, 6.55067420397180027381408133072, 8.157676280752646152128342951143, 8.935952836042187050967796455853, 9.105920592093415705087741316554