| L(s) = 1 | + 2·4-s + 4i·7-s + 6·11-s + i·13-s + 4·16-s + 6i·17-s + 4·19-s − 3i·23-s + 8i·28-s − 3·29-s − 4·31-s − 2i·37-s − 6·41-s − 7i·43-s + 12·44-s + ⋯ |
| L(s) = 1 | + 4-s + 1.51i·7-s + 1.80·11-s + 0.277i·13-s + 16-s + 1.45i·17-s + 0.917·19-s − 0.625i·23-s + 1.51i·28-s − 0.557·29-s − 0.718·31-s − 0.328i·37-s − 0.937·41-s − 1.06i·43-s + 1.80·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.737417114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.737417114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 7iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 9iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.983436881066984547369234034226, −8.205092011179646064713753335775, −7.28319117543009726833182006872, −6.43014381863079405019758494636, −6.05415906160622371351856363184, −5.26216606314948914912926332515, −3.96865150824403263756528565711, −3.23736977879255460318716532151, −2.09627615769300091486610904418, −1.49830945570426003814370886225,
0.905071541249066595623426828020, 1.66202816144099359772524993198, 3.14029777121873409736979204137, 3.67138392430991514183174852451, 4.64399664334264137383902787230, 5.67586533066318116765341780368, 6.58910835641676813175329582130, 7.23921661533166157057491223005, 7.43167987725047131298791554810, 8.582887181217430831922729378782
