| L(s) = 1 | + (1.56 + 1.41i)2-s + (−0.994 + 0.104i)3-s + (0.255 + 2.42i)4-s + (−0.0681 − 2.23i)5-s + (−1.70 − 1.23i)6-s + (2.54 + 0.731i)7-s + (−0.547 + 0.753i)8-s + (0.978 − 0.207i)9-s + (3.04 − 3.59i)10-s + (4.27 + 0.908i)11-s + (−0.507 − 2.38i)12-s + (−3.24 − 1.05i)13-s + (2.94 + 4.73i)14-s + (0.301 + 2.21i)15-s + (2.85 − 0.607i)16-s + (0.693 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (1.10 + 0.997i)2-s + (−0.574 + 0.0603i)3-s + (0.127 + 1.21i)4-s + (−0.0304 − 0.999i)5-s + (−0.696 − 0.505i)6-s + (0.960 + 0.276i)7-s + (−0.193 + 0.266i)8-s + (0.326 − 0.0693i)9-s + (0.963 − 1.13i)10-s + (1.28 + 0.273i)11-s + (−0.146 − 0.689i)12-s + (−0.898 − 0.292i)13-s + (0.788 + 1.26i)14-s + (0.0778 + 0.572i)15-s + (0.714 − 0.151i)16-s + (0.168 + 0.377i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.17884 + 1.09582i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17884 + 1.09582i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.994 - 0.104i)T \) |
| 5 | \( 1 + (0.0681 + 2.23i)T \) |
| 7 | \( 1 + (-2.54 - 0.731i)T \) |
| good | 2 | \( 1 + (-1.56 - 1.41i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (-4.27 - 0.908i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (3.24 + 1.05i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.693 - 1.55i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.0799 - 0.760i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.82 - 1.64i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.565 - 0.410i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.76 - 0.784i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.244 - 1.14i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (2.64 - 8.15i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 13.0iT - 43T^{2} \) |
| 47 | \( 1 + (-1.98 + 4.46i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (11.7 - 1.23i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (3.19 + 3.55i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-3.69 + 4.10i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (3.24 + 7.29i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (11.8 - 8.61i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.49 + 7.03i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-4.11 - 1.83i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (10.0 - 13.8i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-9.97 + 11.0i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (5.33 + 7.34i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.38906123164573685061486693432, −10.06139663229561797294669965726, −9.048152140725310354215060074226, −8.019354390283733534268400733283, −7.17535316398235399375146283231, −6.14583325140909889444029259963, −5.23823694053381828550366513388, −4.72846226291908063175542715112, −3.79870092460120265359037411587, −1.50207603693252003104571547179,
1.55289918567635066078486629419, 2.80923368182636030528424670082, 4.02010598394546412843259071070, 4.76354044923151319439027653203, 5.86233221342184910718642372332, 6.88432394400211908341825354363, 7.78605321411253434644741666707, 9.313187979983183097334832081966, 10.36764828653907813488103901074, 11.06583543207112089087192077421
