L(s) = 1 | + (−0.0623 + 0.0623i)3-s + 0.375i·7-s + 2.99i·9-s + (−2.36 − 2.36i)11-s + (1.76 − 1.76i)13-s + 4.64·17-s + (2.34 − 2.34i)19-s + (−0.0234 − 0.0234i)21-s + 2.07i·23-s + (−0.373 − 0.373i)27-s + (2.55 − 2.55i)29-s − 8.51·31-s + 0.295·33-s + (7.62 + 7.62i)37-s + 0.219i·39-s + ⋯ |
L(s) = 1 | + (−0.0359 + 0.0359i)3-s + 0.142i·7-s + 0.997i·9-s + (−0.713 − 0.713i)11-s + (0.489 − 0.489i)13-s + 1.12·17-s + (0.539 − 0.539i)19-s + (−0.00511 − 0.00511i)21-s + 0.433i·23-s + (−0.0718 − 0.0718i)27-s + (0.474 − 0.474i)29-s − 1.52·31-s + 0.0513·33-s + (1.25 + 1.25i)37-s + 0.0352i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.684635776\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.684635776\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.0623 - 0.0623i)T - 3iT^{2} \) |
| 7 | \( 1 - 0.375iT - 7T^{2} \) |
| 11 | \( 1 + (2.36 + 2.36i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.76 + 1.76i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.64T + 17T^{2} \) |
| 19 | \( 1 + (-2.34 + 2.34i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.07iT - 23T^{2} \) |
| 29 | \( 1 + (-2.55 + 2.55i)T - 29iT^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + (-7.62 - 7.62i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.77iT - 41T^{2} \) |
| 43 | \( 1 + (-6.21 - 6.21i)T + 43iT^{2} \) |
| 47 | \( 1 - 9.71T + 47T^{2} \) |
| 53 | \( 1 + (3.03 + 3.03i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.11 - 8.11i)T + 59iT^{2} \) |
| 61 | \( 1 + (-0.728 + 0.728i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.969 - 0.969i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.14iT - 71T^{2} \) |
| 73 | \( 1 - 7.56iT - 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + (-10.6 + 10.6i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.7iT - 89T^{2} \) |
| 97 | \( 1 - 3.86T + 97T^{2} \) |
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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
−9.509662743390885008643558822775, −8.516721463578341950204809766895, −7.86069956214989645311278256250, −7.29525042769882342439060740503, −5.88607400540423360485260463927, −5.52787888733134158665996624000, −4.53412644066046931124186191066, −3.31317057086831536784665477712, −2.51917350281584931454119406604, −1.01712954267681618746039681409,
0.869748646613034443958221148599, 2.23001939534301278978232870453, 3.48402884389886278930202862067, 4.17463740654590699036890925160, 5.42104004151504213845855628622, 5.99670109620296330067582587160, 7.17324863482233680691377922472, 7.54390715759414423381386863805, 8.709777367146971883956950026678, 9.339261827966054990073752475728