L(s) = 1 | + (−0.707 + 0.707i)2-s + (−0.707 + 0.707i)3-s − 1.00i·4-s + (0.461 − 2.18i)5-s − 1.00i·6-s + (0.707 + 0.707i)8-s − 1.00i·9-s + (1.22 + 1.87i)10-s − 2.98·11-s + (0.707 + 0.707i)12-s + (−0.960 + 0.960i)13-s + (1.22 + 1.87i)15-s − 1.00·16-s + (−1.62 − 1.62i)17-s + (0.707 + 0.707i)18-s + 8.67·19-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (−0.408 + 0.408i)3-s − 0.500i·4-s + (0.206 − 0.978i)5-s − 0.408i·6-s + (0.250 + 0.250i)8-s − 0.333i·9-s + (0.385 + 0.592i)10-s − 0.899·11-s + (0.204 + 0.204i)12-s + (−0.266 + 0.266i)13-s + (0.315 + 0.483i)15-s − 0.250·16-s + (−0.394 − 0.394i)17-s + (0.166 + 0.166i)18-s + 1.98·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4749893396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4749893396\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.461 + 2.18i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 2.98T + 11T^{2} \) |
| 13 | \( 1 + (0.960 - 0.960i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.62 + 1.62i)T + 17iT^{2} \) |
| 19 | \( 1 - 8.67T + 19T^{2} \) |
| 23 | \( 1 + (-1.36 - 1.36i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.00iT - 29T^{2} \) |
| 31 | \( 1 + 0.179iT - 31T^{2} \) |
| 37 | \( 1 + (4.86 - 4.86i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.14iT - 41T^{2} \) |
| 43 | \( 1 + (7.01 + 7.01i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.202 - 0.202i)T + 47iT^{2} \) |
| 53 | \( 1 + (7.01 + 7.01i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 + 2.41iT - 61T^{2} \) |
| 67 | \( 1 + (6.29 - 6.29i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.08T + 71T^{2} \) |
| 73 | \( 1 + (8.78 - 8.78i)T - 73iT^{2} \) |
| 79 | \( 1 + 16.2iT - 79T^{2} \) |
| 83 | \( 1 + (-8.11 + 8.11i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 + (-7.44 - 7.44i)T + 97iT^{2} \) |
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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
−9.290043716010047046205633037558, −8.560063910631368598060637591167, −7.67246945486024083120526500143, −6.95703160331051175072458271695, −5.75812053830462583663203862397, −5.21066512370421166720321395060, −4.57291448363719071077750085292, −3.15782824628179088702868965640, −1.62334109794843661474721453982, −0.24092409628550973267663530484,
1.44891190248953267679651567613, 2.68370379698863319864645552367, 3.31960584419453372403274072222, 4.81014766791964966200606753208, 5.71400716031413787959689353109, 6.65409428101839260309547922167, 7.49242974771367992072408173405, 7.917641887110297045947388214100, 9.123515916078323181395069198016, 9.916205598794640671942418053740