L(s) = 1 | + 3i·3-s + 4i·5-s + 7-s − 6·9-s − 5i·13-s − 12·15-s − 5·17-s + i·19-s + 3i·21-s − 3·23-s − 11·25-s − 9i·27-s + 7i·29-s + 10·31-s + 4i·35-s + ⋯ |
L(s) = 1 | + 1.73i·3-s + 1.78i·5-s + 0.377·7-s − 2·9-s − 1.38i·13-s − 3.09·15-s − 1.21·17-s + 0.229i·19-s + 0.654i·21-s − 0.625·23-s − 2.20·25-s − 1.73i·27-s + 1.29i·29-s + 1.79·31-s + 0.676i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.101665486\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101665486\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - iT \) |
good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 5 | \( 1 - 4iT - 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 7iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 - iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−10.34162321963084756228495507212, −9.789944851281245100404629534036, −8.702666780100205789920154436905, −7.931113381001288443833542932298, −6.82861611084092418158079627366, −5.97086886665497760988066145289, −5.06095169685955457986079072353, −4.07736104185211590370685136133, −3.23270879701863519368115653074, −2.55192185855878262092093367963,
0.45615319848583992520295124011, 1.59768397922571576498652599915, 2.25676566650142583540662821214, 4.23343298498192265577536023043, 4.92058352265273508167847449905, 6.09620023331127049825297063168, 6.66782394428390095259465866418, 7.74981057464855372059181133882, 8.349699421063129414052767235301, 8.888684763342553795967179645308