L(s) = 1 | − 2-s + (0.707 − 0.707i)3-s + 4-s + (−1.64 + 1.51i)5-s + (−0.707 + 0.707i)6-s + (2.75 − 2.75i)7-s − 8-s − 1.00i·9-s + (1.64 − 1.51i)10-s + 0.350i·11-s + (0.707 − 0.707i)12-s + 3.27·13-s + (−2.75 + 2.75i)14-s + (−0.0966 + 2.23i)15-s + 16-s + 7.07i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.408 − 0.408i)3-s + 0.5·4-s + (−0.737 + 0.675i)5-s + (−0.288 + 0.288i)6-s + (1.04 − 1.04i)7-s − 0.353·8-s − 0.333i·9-s + (0.521 − 0.477i)10-s + 0.105i·11-s + (0.204 − 0.204i)12-s + 0.907·13-s + (−0.737 + 0.737i)14-s + (−0.0249 + 0.576i)15-s + 0.250·16-s + 1.71i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.388918135\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.388918135\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.64 - 1.51i)T \) |
| 37 | \( 1 + (3.58 - 4.91i)T \) |
good | 7 | \( 1 + (-2.75 + 2.75i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.350iT - 11T^{2} \) |
| 13 | \( 1 - 3.27T + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 + (-2.01 - 2.01i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + (-6.14 + 6.14i)T - 29iT^{2} \) |
| 31 | \( 1 + (2.76 + 2.76i)T + 31iT^{2} \) |
| 41 | \( 1 + 5.57iT - 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + (-5.76 + 5.76i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.45 + 3.45i)T + 53iT^{2} \) |
| 59 | \( 1 + (-10.6 - 10.6i)T + 59iT^{2} \) |
| 61 | \( 1 + (-1.30 - 1.30i)T + 61iT^{2} \) |
| 67 | \( 1 + (-10.6 - 10.6i)T + 67iT^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + (-0.998 + 0.998i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.47 - 2.47i)T + 79iT^{2} \) |
| 83 | \( 1 + (10.6 + 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.03 + 3.03i)T - 89iT^{2} \) |
| 97 | \( 1 + 7.94iT - 97T^{2} \) |
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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
−10.04394834146959566117911750082, −8.458622496867004947228046066178, −8.291173708759761342700917747488, −7.48273267723817065135198618751, −6.79399338241964978614796363575, −5.83464438597716687909234607964, −4.16641939195127968511719185452, −3.63413813436463853562615600101, −2.13941173763256503041873268226, −1.00493091243264479846170757151,
1.07312420925853396853198351342, 2.45763796710747652149893370318, 3.55746438986742960951815939167, 4.89503427232038167874981452448, 5.33088661444227240412864193744, 6.79081814829835070451903360747, 7.80514187074968380110209268526, 8.360787997459584819865953152209, 9.044173539903431537151847804689, 9.469134366393036335586733273378