L(s) = 1 | + i·3-s + 2·9-s + 3·11-s − i·13-s + 3i·17-s + 2·19-s + 6i·23-s + 5i·27-s + 9·29-s − 8·31-s + 3i·33-s − 10i·37-s + 39-s − 2i·43-s + 3i·47-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.666·9-s + 0.904·11-s − 0.277i·13-s + 0.727i·17-s + 0.458·19-s + 1.25i·23-s + 0.962i·27-s + 1.67·29-s − 1.43·31-s + 0.522i·33-s − 1.64i·37-s + 0.160·39-s − 0.304i·43-s + 0.437i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.272284336\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.272284336\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−8.603065637268299122701240434488, −7.46597890896270320179846014565, −7.14171544229647254250842592286, −6.10849222096789831357377938109, −5.48197429969527756623327698934, −4.58421121154922232442410868444, −3.86818284312384878420606600200, −3.31651143966793661020085412698, −1.98689270840773951893061333010, −1.06127112404737090895035348434,
0.74354561630153336456397729461, 1.62176828910906757940563330815, 2.60884281419296112573075834363, 3.58985731513789591328571091730, 4.49710170317929390161925092859, 5.07036069492517075456504060293, 6.33546966474839103338400339414, 6.62033696800988169340629896352, 7.32974718307803183203203355325, 8.067752254235522766690821442662