L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + 0.585i·11-s + (−3.43 + 3.43i)13-s − 1.00·14-s − 1.00·16-s + (−0.906 + 0.906i)17-s − 2.04i·19-s + (0.414 + 0.414i)22-s + (0.257 + 0.257i)23-s + 4.86i·26-s + (−0.707 + 0.707i)28-s − 1.75·29-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + 0.176i·11-s + (−0.953 + 0.953i)13-s − 0.267·14-s − 0.250·16-s + (−0.219 + 0.219i)17-s − 0.470i·19-s + (0.0883 + 0.0883i)22-s + (0.0537 + 0.0537i)23-s + 0.953i·26-s + (−0.133 + 0.133i)28-s − 0.325·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.096233614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.096233614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 - 0.585iT - 11T^{2} \) |
| 13 | \( 1 + (3.43 - 3.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.906 - 0.906i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.04iT - 19T^{2} \) |
| 23 | \( 1 + (-0.257 - 0.257i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.75T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 + (-4.04 - 4.04i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.84iT - 41T^{2} \) |
| 43 | \( 1 + (5.10 - 5.10i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.49 - 5.49i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.02 - 5.02i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 + (-6.74 - 6.74i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.9iT - 71T^{2} \) |
| 73 | \( 1 + (5.97 - 5.97i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.944iT - 79T^{2} \) |
| 83 | \( 1 + (7.82 + 7.82i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.0705T + 89T^{2} \) |
| 97 | \( 1 + (-0.746 - 0.746i)T + 97iT^{2} \) |
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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
−9.026103882644742948398513342087, −8.080398186270821110309168092175, −7.14908794724127823751689936236, −6.59395663593521443063765295407, −5.73715157121825525973533205822, −4.67711194220634892470610986104, −4.33195201456855197116267200284, −3.18343246105206921119910253007, −2.39398410414490926861789224403, −1.31406883418170422371505121307,
0.27727964515687568696094245146, 2.07574004074383983011991093315, 3.02114468244778674067375509310, 3.80347974796809163570093542866, 4.86897179400169700531312231062, 5.44205613624207478594227526379, 6.20460807369489042421458092006, 7.01381045866067383472472443405, 7.70733148821243876487254296325, 8.364677940403353133371390620111