| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − i·7-s + i·8-s − 9-s − 3·11-s + i·12-s − 2i·13-s − 14-s + 16-s − i·17-s + i·18-s + 7·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.288i·12-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s − 0.242i·17-s + 0.235i·18-s + 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5182773127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5182773127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + iT \) |
| good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 7T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 5iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 17T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.395875628819618426841910050780, −7.52431935486464251488106197899, −7.18922747516542450082361005697, −5.80541278797801446915087566900, −5.32972968541709628887106147473, −4.26653256048889989246882328442, −3.22297936132339876477210583767, −2.51957767955154180736522142661, −1.33781476584183833567644085153, −0.17387535493892419288918454133,
1.72632459164441281525344687421, 3.13277066085658569817202219749, 3.82056319352285405482983521168, 5.07838740298404740682706793087, 5.34506079162002874048220604551, 6.21514010011937852838357450134, 7.34587201565838584997364620744, 7.69990400716311455162435486876, 8.756441586066739600777039885811, 9.279353336671326402025373360324
