| L(s) = 1 | − 41.8i·2-s + 0.232i·3-s − 1.23e3·4-s + 9.71·6-s − 2.40e3i·7-s + 3.02e4i·8-s + 1.96e4·9-s + 1.74e4·11-s − 287. i·12-s − 1.22e5i·13-s − 1.00e5·14-s + 6.31e5·16-s − 3.31e5i·17-s − 8.22e5i·18-s − 7.61e5·19-s + ⋯ |
| L(s) = 1 | − 1.84i·2-s + 0.00165i·3-s − 2.41·4-s + 0.00305·6-s − 0.377i·7-s + 2.61i·8-s + 0.999·9-s + 0.358·11-s − 0.00399i·12-s − 1.18i·13-s − 0.698·14-s + 2.40·16-s − 0.963i·17-s − 1.84i·18-s − 1.34·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2349859787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2349859787\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 2.40e3iT \) |
| good | 2 | \( 1 + 41.8iT - 512T^{2} \) |
| 3 | \( 1 - 0.232iT - 1.96e4T^{2} \) |
| 11 | \( 1 - 1.74e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.22e5iT - 1.06e10T^{2} \) |
| 17 | \( 1 + 3.31e5iT - 1.18e11T^{2} \) |
| 19 | \( 1 + 7.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.23e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 6.34e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.03e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 7.37e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.06e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 2.03e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + 5.97e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 + 6.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.44e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.19e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 5.58e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.54e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 4.51e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.34e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 6.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.42e9iT - 7.60e17T^{2} \) |
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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
−10.26938908899253293883309211308, −9.574122841036729857225694060983, −8.476867938468975090877531533661, −7.16254627019014735080741584377, −5.32392643963797378463785601067, −4.21189683884799826338780450399, −3.36463948561674419248831347104, −2.09522946859034318198371959893, −1.07632412304607254020876303177, −0.05888862680510438162117179853,
1.69670538405870065630962363110, 4.02113796465432456286662043056, 4.67009652998445790268482453126, 6.12480597037519582849115107964, 6.66319043224519268837625477050, 7.71117019110548800070389249248, 8.747124952924361580484625026152, 9.420467794446774905005740129243, 10.70113266286014843337724186097, 12.39771453510329306207291881941
