| L(s) = 1 | + (−2 − 2i)3-s + 5i·9-s + (−2 + 2i)11-s + 2·13-s + (−4 − i)17-s + (4 − 4i)23-s + (4 − 4i)27-s + (5 + 5i)29-s + (4 + 4i)31-s + 8·33-s + (5 + 5i)37-s + (−4 − 4i)39-s + (−5 + 5i)41-s − 4i·43-s + 8·47-s + ⋯ |
| L(s) = 1 | + (−1.15 − 1.15i)3-s + 1.66i·9-s + (−0.603 + 0.603i)11-s + 0.554·13-s + (−0.970 − 0.242i)17-s + (0.834 − 0.834i)23-s + (0.769 − 0.769i)27-s + (0.928 + 0.928i)29-s + (0.718 + 0.718i)31-s + 1.39·33-s + (0.821 + 0.821i)37-s + (−0.640 − 0.640i)39-s + (−0.780 + 0.780i)41-s − 0.609i·43-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9768131694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9768131694\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (4 + i)T \) |
| good | 3 | \( 1 + (2 + 2i)T + 3iT^{2} \) |
| 7 | \( 1 - 7iT^{2} \) |
| 11 | \( 1 + (2 - 2i)T - 11iT^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (-4 + 4i)T - 23iT^{2} \) |
| 29 | \( 1 + (-5 - 5i)T + 29iT^{2} \) |
| 31 | \( 1 + (-4 - 4i)T + 31iT^{2} \) |
| 37 | \( 1 + (-5 - 5i)T + 37iT^{2} \) |
| 41 | \( 1 + (5 - 5i)T - 41iT^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + (-3 + 3i)T - 61iT^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + (8 + 8i)T + 71iT^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 + (-12 + 12i)T - 79iT^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 + (5 + 5i)T + 97iT^{2} \) |
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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
−9.104446984880482535451294567535, −8.271505505873663968927078125636, −7.45319625946297801025026681457, −6.60749344087152747420305580709, −6.34152533672839124101820142497, −5.09942748690817642374122429448, −4.65543444304814713876824697850, −2.98339131850897549821335461615, −1.86097300501943037655808360965, −0.71323058688322813349750771031,
0.73305218182036777496926018034, 2.62679153113629835222122067614, 3.85969261507028677148406802453, 4.48277300666596239312247168959, 5.45428777403582425152781298823, 5.95731205397230811337566372247, 6.81236189634390888074544023495, 7.964425256306221473386442904012, 8.873801745127640823346103879134, 9.547518841460469826325183626295
