| L(s) = 1 | + i·2-s + 4-s − i·7-s + 3i·8-s − 6i·13-s + 14-s − 16-s + 2i·17-s + 8·19-s − 8i·23-s + 6·26-s − i·28-s − 2·29-s + 4·31-s + 5i·32-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.5·4-s − 0.377i·7-s + 1.06i·8-s − 1.66i·13-s + 0.267·14-s − 0.250·16-s + 0.485i·17-s + 1.83·19-s − 1.66i·23-s + 1.17·26-s − 0.188i·28-s − 0.371·29-s + 0.718·31-s + 0.883i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141030480\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141030480\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 2 | \( 1 - iT - 2T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−9.490983113544726911096892281442, −8.232235082359792221382278554871, −7.942708496971895126372494898041, −7.10458527971566872573636769625, −6.26943173189725074943512818816, −5.54589314464760391558465856111, −4.74574884356674026826544923732, −3.36565902875426010678441343639, −2.55451499972035659248183105405, −0.975673123889942811275946981400,
1.23092617630934680890913793523, 2.20307336360287510509149123793, 3.21345964266867327788660196480, 4.06139191848469787851416503012, 5.24483514627028430888491547916, 6.10638354132874879162814788058, 7.15692283948592883349122970358, 7.48858141921298755786638729116, 8.894731256731710977761461520416, 9.515463804908827023448684501502
