L(s) = 1 | + (0.707 − 0.707i)5-s − 0.841·7-s + (−3.43 − 3.43i)11-s + (−0.690 + 0.690i)13-s − 0.821i·17-s + (3.35 + 3.35i)19-s − 0.473i·23-s − 1.00i·25-s + (0.820 + 0.820i)29-s − 2.43i·31-s + (−0.595 + 0.595i)35-s + (−6.22 − 6.22i)37-s + 3.45·41-s + (−3.26 + 3.26i)43-s − 0.936·47-s + ⋯ |
L(s) = 1 | + (0.316 − 0.316i)5-s − 0.318·7-s + (−1.03 − 1.03i)11-s + (−0.191 + 0.191i)13-s − 0.199i·17-s + (0.770 + 0.770i)19-s − 0.0987i·23-s − 0.200i·25-s + (0.152 + 0.152i)29-s − 0.436i·31-s + (−0.100 + 0.100i)35-s + (−1.02 − 1.02i)37-s + 0.539·41-s + (−0.497 + 0.497i)43-s − 0.136·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3809993252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3809993252\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + 0.841T + 7T^{2} \) |
| 11 | \( 1 + (3.43 + 3.43i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.690 - 0.690i)T - 13iT^{2} \) |
| 17 | \( 1 + 0.821iT - 17T^{2} \) |
| 19 | \( 1 + (-3.35 - 3.35i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.473iT - 23T^{2} \) |
| 29 | \( 1 + (-0.820 - 0.820i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.43iT - 31T^{2} \) |
| 37 | \( 1 + (6.22 + 6.22i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.45T + 41T^{2} \) |
| 43 | \( 1 + (3.26 - 3.26i)T - 43iT^{2} \) |
| 47 | \( 1 + 0.936T + 47T^{2} \) |
| 53 | \( 1 + (6.73 - 6.73i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.53 + 9.53i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.32 - 6.32i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.77 + 2.77i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 - 6.65iT - 73T^{2} \) |
| 79 | \( 1 - 15.9iT - 79T^{2} \) |
| 83 | \( 1 + (-3.05 + 3.05i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.274742396105195138681270145437, −7.84028679777475627327549910490, −6.87874214537732601460005999280, −5.94777387679447910244317781008, −5.44877030312066433635751341733, −4.57439548054251422473613181480, −3.44227700196239882855986220924, −2.73648807376118306070023395170, −1.49960852542867843332544996278, −0.11492100718968963791404772103,
1.60679277037255733506075669784, 2.66677031572453399508446395296, 3.34876780796091611832277926610, 4.67670412876401720585071001129, 5.15902061348293300700914230976, 6.14200140835583410930296884885, 6.95761137823905180933092312831, 7.52758482451894951246246940301, 8.321848417314580199085346498919, 9.277289463629860350589423897591