L(s) = 1 | − i·2-s + 4-s + (2 − i)5-s + i·7-s − 3i·8-s + (−1 − 2i)10-s + 4i·13-s + 14-s − 16-s − 2i·17-s + (2 − i)20-s + (3 − 4i)25-s + 4·26-s + i·28-s − 8·29-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.5·4-s + (0.894 − 0.447i)5-s + 0.377i·7-s − 1.06i·8-s + (−0.316 − 0.632i)10-s + 1.10i·13-s + 0.267·14-s − 0.250·16-s − 0.485i·17-s + (0.447 − 0.223i)20-s + (0.600 − 0.800i)25-s + 0.784·26-s + 0.188i·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.46104 - 0.902978i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.46104 - 0.902978i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−11.41298623853975061685759693959, −10.75727729891265546718330705044, −9.430266607449692733954377527046, −9.262833880362840912141061639575, −7.60552726237620616303815788374, −6.51252068478677711981942667357, −5.57982835795838988860740791426, −4.17442334522816193409836691426, −2.63104728246724548777217729413, −1.59045257348155489523940899471,
1.97495817733980094558940203269, 3.36057546956158117358363313220, 5.25729705408890094718010845592, 5.98038831215363544359585799797, 6.96621511707097220402507385213, 7.76025142873707417305208124346, 8.871821485550934700702754864555, 10.15956707185444641380303316903, 10.70592198453729670682339742731, 11.70176983766197110931519177941