| L(s) = 1 | − i·2-s + 3-s − 4-s − i·6-s − 2.60i·7-s + i·8-s + 9-s − 12-s + 3.60·13-s − 2.60·14-s + 16-s + 2.60·17-s − i·18-s + 2.60i·19-s − 2.60i·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.408i·6-s − 0.984i·7-s + 0.353i·8-s + 0.333·9-s − 0.288·12-s + 1.00·13-s − 0.696·14-s + 0.250·16-s + 0.631·17-s − 0.235i·18-s + 0.597i·19-s − 0.568i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.202881773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.202881773\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 3.60T \) |
| good | 7 | \( 1 + 2.60iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 - 2.60iT - 19T^{2} \) |
| 23 | \( 1 - 8.60T + 23T^{2} \) |
| 29 | \( 1 + 2.60T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 + 5.21iT - 37T^{2} \) |
| 41 | \( 1 + 11.2iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 5.21iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 5.21iT - 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 + 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 5.21iT - 71T^{2} \) |
| 73 | \( 1 - 8.60iT - 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 17.2iT - 83T^{2} \) |
| 89 | \( 1 - 0.788iT - 89T^{2} \) |
| 97 | \( 1 + 8.60iT - 97T^{2} \) |
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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
−8.996487614434031503973700183865, −8.416432441332995297608687676959, −7.48025266234818845812003289320, −6.84312221597269205843797275719, −5.63489092925202566628667700759, −4.70754746366557470192239716234, −3.62652705167760180808300020464, −3.32477372344949089044051845598, −1.87123463342331502262106039997, −0.891295285628353587483195737751,
1.25453944220236276667067501143, 2.69830541979073193496203521080, 3.48752189985253012042042503852, 4.65891843262147813886325007731, 5.42604272272907554319016985050, 6.29011593065188401812238925527, 6.98629796963402494981201305404, 8.038475175146344909042752031929, 8.436262626808759713260087369128, 9.311668729835843824220749238574
