| 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
−8.863138123097766565097026748424, −7.64665204246027160715846737290, −7.03758745361528793383739734931, −6.22835169981652046745357899188, −5.37387626185654187098060303229, −4.61606017107827800766223858492, −3.74368257497584441171611306751, −2.80233672930197023304008225621, −2.05301591888127069271447865275, −0.48267235658192485236899349974,
1.69218391416249308417891512628, 2.81747109707590448963583575948, 3.79221575903291430886022969184, 4.58849502344191835176731022708, 5.44491485423459601849545928914, 6.00303460066382348770266106700, 6.89518872194911742834909093832, 7.80372664834874397402118172274, 8.279913215350565753826534963523, 9.082597970095303739292313640560
