L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.577 − 2.16i)5-s + (−0.575 − 0.575i)7-s + 1.00i·9-s − 2.55i·11-s + (−1.98 − 1.98i)13-s + (1.11 − 1.93i)15-s + (−1.42 − 1.42i)17-s − 3.58·19-s − 0.813i·21-s + (0.644 + 4.75i)23-s + (−4.33 + 2.49i)25-s + (−0.707 + 0.707i)27-s + 3.90i·29-s − 7.98·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.258 − 0.966i)5-s + (−0.217 − 0.217i)7-s + 0.333i·9-s − 0.771i·11-s + (−0.549 − 0.549i)13-s + (0.289 − 0.499i)15-s + (−0.344 − 0.344i)17-s − 0.823·19-s − 0.177i·21-s + (0.134 + 0.990i)23-s + (−0.866 + 0.498i)25-s + (−0.136 + 0.136i)27-s + 0.725i·29-s − 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7502351396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7502351396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.577 + 2.16i)T \) |
| 23 | \( 1 + (-0.644 - 4.75i)T \) |
good | 7 | \( 1 + (0.575 + 0.575i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.55iT - 11T^{2} \) |
| 13 | \( 1 + (1.98 + 1.98i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.42 + 1.42i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.58T + 19T^{2} \) |
| 29 | \( 1 - 3.90iT - 29T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 + (5.07 + 5.07i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.13T + 41T^{2} \) |
| 43 | \( 1 + (-5.16 + 5.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.115 - 0.115i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.73 - 2.73i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.87iT - 59T^{2} \) |
| 61 | \( 1 + 5.54iT - 61T^{2} \) |
| 67 | \( 1 + (-5.83 - 5.83i)T + 67iT^{2} \) |
| 71 | \( 1 - 0.976T + 71T^{2} \) |
| 73 | \( 1 + (0.307 + 0.307i)T + 73iT^{2} \) |
| 79 | \( 1 + 6.52T + 79T^{2} \) |
| 83 | \( 1 + (-4.68 + 4.68i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.06T + 89T^{2} \) |
| 97 | \( 1 + (5.38 + 5.38i)T + 97iT^{2} \) |
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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
−9.078850847269915674525558871654, −8.661988625011749188481309248428, −7.75469511055562066043923185743, −6.98725271094011842543630936746, −5.64265197482246205974864541436, −5.08994057182850803495217938726, −4.00028061482123712605040392342, −3.28362134517895202433612789819, −1.88464069689602280800777824580, −0.26770637044222489935722331317,
1.94837472042817883220769722343, 2.67255844010039310721971805424, 3.81993960933419763772359600139, 4.68847917687529453804652093650, 6.08724653424678541111429922648, 6.75147160451133373265654745454, 7.36353011025419921520152938598, 8.221807598794432931878020615366, 9.086841112528654750066407756159, 9.891291106012851659460428300989