L(s) = 1 | + (0.905 + 1.77i)2-s + (−1.74 + 2.40i)4-s + (−0.233 − 0.972i)5-s + (−3.89 − 0.616i)8-s + (1.51 − 1.29i)10-s + (−1.62 − 0.526i)11-s + (−0.891 − 0.453i)13-s + (−1.50 − 4.63i)16-s + (2.74 + 1.13i)20-s + (−0.531 − 3.35i)22-s + (−0.891 + 0.453i)25-s − 1.99i·26-s + (4.09 − 4.09i)32-s + (0.309 + 3.92i)40-s + (−1.23 + 0.401i)41-s + ⋯ |
L(s) = 1 | + (0.905 + 1.77i)2-s + (−1.74 + 2.40i)4-s + (−0.233 − 0.972i)5-s + (−3.89 − 0.616i)8-s + (1.51 − 1.29i)10-s + (−1.62 − 0.526i)11-s + (−0.891 − 0.453i)13-s + (−1.50 − 4.63i)16-s + (2.74 + 1.13i)20-s + (−0.531 − 3.35i)22-s + (−0.891 + 0.453i)25-s − 1.99i·26-s + (4.09 − 4.09i)32-s + (0.309 + 3.92i)40-s + (−1.23 + 0.401i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2226439225\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2226439225\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.233 + 0.972i)T \) |
| 13 | \( 1 + (0.891 + 0.453i)T \) |
good | 2 | \( 1 + (-0.905 - 1.77i)T + (-0.587 + 0.809i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (1.62 + 0.526i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (1.23 - 0.401i)T + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (-0.642 + 0.642i)T - iT^{2} \) |
| 47 | \( 1 + (1.28 - 0.203i)T + (0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.600 + 1.84i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (-0.614 + 0.845i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.154 - 0.0245i)T + (0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (0.236 - 0.727i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.951 - 0.309i)T^{2} \) |
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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
−8.332084192091379749183330671827, −8.003819804149616723943527574850, −7.44091095017712295489552831183, −6.50283435487640379470182577638, −5.59195373013506493920272805443, −5.12945510369679493798898042734, −4.63084232394201052232006572692, −3.57706382207566834066219499331, −2.71161796146616772275271066422, −0.094508845380614241131495974781,
1.85983452655965294572959895420, 2.58934752600722832664246735293, 3.16853950897692591180493397750, 4.17003907704953877744905227573, 4.90958269398491980052431552103, 5.55621202099412222805427415151, 6.54533148470268522708289179825, 7.48315815952309111948370010063, 8.491022002358227266480390559608, 9.511405350825788017124938245361