L(s) = 1 | + i·7-s − 6·11-s + 5i·13-s − 6i·17-s − 5·19-s + 6i·23-s − 6·29-s − 31-s − 2i·37-s − i·43-s + 6i·47-s + 6·49-s + 12i·53-s − 6·59-s − 13·61-s + ⋯ |
L(s) = 1 | + 0.377i·7-s − 1.80·11-s + 1.38i·13-s − 1.45i·17-s − 1.14·19-s + 1.25i·23-s − 1.11·29-s − 0.179·31-s − 0.328i·37-s − 0.152i·43-s + 0.875i·47-s + 0.857·49-s + 1.64i·53-s − 0.781·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.101794 + 0.431210i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.101794 + 0.431210i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 6T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−10.57035585260762501504220656045, −9.435590974838287656801479702750, −9.009248093768973114603293531539, −7.77257173406062130709802700467, −7.26442280926462827992290063716, −6.06321439432679930640564944032, −5.20963005992701011660948774043, −4.33422707656086795869755599092, −2.94284828645428034475902093026, −1.98586538432933420897346775636,
0.19230769664162608861973081968, 2.12064936219419524615740731141, 3.22525608976952906489068533611, 4.38823556100248134338029319870, 5.41916014624861283158612246916, 6.16490166397990327785233876558, 7.36767190819819267159735340736, 8.127989041971554567784853107520, 8.643413251964296609821545157696, 10.20148949022245829906742717178