L(s) = 1 | + i·7-s + 3·11-s + 4i·13-s + 6i·17-s + 4·19-s + 3i·23-s + 3·29-s − 10·31-s − 7i·37-s + i·43-s + 12i·47-s − 49-s + 6i·53-s + 12·59-s − 4·61-s + ⋯ |
L(s) = 1 | + 0.377i·7-s + 0.904·11-s + 1.10i·13-s + 1.45i·17-s + 0.917·19-s + 0.625i·23-s + 0.557·29-s − 1.79·31-s − 1.15i·37-s + 0.152i·43-s + 1.75i·47-s − 0.142·49-s + 0.824i·53-s + 1.56·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.637929853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.637929853\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.322096589156685691598145824695, −7.46427488600430883842200283114, −6.92111606218338561244311284497, −6.01781934009883028351677982200, −5.64973004598675539874019358222, −4.51058905661888656209865764273, −3.93222993704373413354317890318, −3.15503970886952016409268439764, −1.95178073243425631827428459389, −1.33645845069426978825235813149,
0.42757675158975702221547456367, 1.35017779617208586007762528395, 2.60224800943285292542151671419, 3.34950228022828918976509199505, 4.08762263087621523526595477272, 5.10196232464633745481516783849, 5.49030598156504651563292690032, 6.56603745670835960160324422235, 7.11560338521717918413242425983, 7.68519459455951485668326698699