| L(s) = 1 | + 2-s + i·3-s + 4-s + i·6-s − 2·7-s + 8-s − 9-s + i·12-s + (2 − 3i)13-s − 2·14-s + 16-s + 2i·17-s − 18-s + 6i·19-s − 2i·21-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s − 0.755·7-s + 0.353·8-s − 0.333·9-s + 0.288i·12-s + (0.554 − 0.832i)13-s − 0.534·14-s + 0.250·16-s + 0.485i·17-s − 0.235·18-s + 1.37i·19-s − 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.356625217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356625217\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-2 + 3i)T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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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
−9.694016712378035754981064154288, −8.366838030368023499126782718250, −7.997860966368313205547891046796, −6.63267108306935304094952216906, −6.16902622843767545765323769256, −5.29884684416494913216419414403, −4.45720548217776739254632501244, −3.39699469984014380832398708088, −3.04254088241409638182579670218, −1.39269502289205923453683145862,
0.69984946794002142280064562161, 2.25701906041595830660196857955, 2.96608493158294955620640159852, 4.11468861395812635048779299946, 4.86735091020590884769521277253, 6.02779769269554767986543824390, 6.55678940095100445499472169153, 7.14150648635048845759919357393, 8.137369780441600156200967963196, 8.998056782952174325958252430631
