| L(s) = 1 | + (1.17 + 0.788i)2-s + (0.756 + 1.85i)4-s − i·5-s + 0.625i·7-s + (−0.570 + 2.77i)8-s + (0.788 − 1.17i)10-s − 1.38·11-s − 13-s + (−0.492 + 0.733i)14-s + (−2.85 + 2.80i)16-s + 1.76i·17-s + 5.22i·19-s + (1.85 − 0.756i)20-s + (−1.63 − 1.09i)22-s − 2.64·23-s + ⋯ |
| L(s) = 1 | + (0.830 + 0.557i)2-s + (0.378 + 0.925i)4-s − 0.447i·5-s + 0.236i·7-s + (−0.201 + 0.979i)8-s + (0.249 − 0.371i)10-s − 0.418·11-s − 0.277·13-s + (−0.131 + 0.196i)14-s + (−0.713 + 0.700i)16-s + 0.428i·17-s + 1.19i·19-s + (0.413 − 0.169i)20-s + (−0.347 − 0.233i)22-s − 0.552·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.038035969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.038035969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.17 - 0.788i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 0.625iT - 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 17 | \( 1 - 1.76iT - 17T^{2} \) |
| 19 | \( 1 - 5.22iT - 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 - 0.138iT - 29T^{2} \) |
| 31 | \( 1 - 5.31iT - 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 2.26iT - 41T^{2} \) |
| 43 | \( 1 - 4.50iT - 43T^{2} \) |
| 47 | \( 1 + 0.339T + 47T^{2} \) |
| 53 | \( 1 - 4.58iT - 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 - 0.661iT - 67T^{2} \) |
| 71 | \( 1 - 1.23T + 71T^{2} \) |
| 73 | \( 1 - 4.49T + 73T^{2} \) |
| 79 | \( 1 - 13.2iT - 79T^{2} \) |
| 83 | \( 1 + 3.37T + 83T^{2} \) |
| 89 | \( 1 + 5.94iT - 89T^{2} \) |
| 97 | \( 1 + 3.98T + 97T^{2} \) |
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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
−9.082597970095303739292313640560, −8.279913215350565753826534963523, −7.80372664834874397402118172274, −6.89518872194911742834909093832, −6.00303460066382348770266106700, −5.44491485423459601849545928914, −4.58849502344191835176731022708, −3.79221575903291430886022969184, −2.81747109707590448963583575948, −1.69218391416249308417891512628,
0.48267235658192485236899349974, 2.05301591888127069271447865275, 2.80233672930197023304008225621, 3.74368257497584441171611306751, 4.61606017107827800766223858492, 5.37387626185654187098060303229, 6.22835169981652046745357899188, 7.03758745361528793383739734931, 7.64665204246027160715846737290, 8.863138123097766565097026748424
