L(s) = 1 | − 2-s + (0.707 + 0.707i)3-s + 4-s + (2.05 − 0.887i)5-s + (−0.707 − 0.707i)6-s + (−1.84 − 1.84i)7-s − 8-s + 1.00i·9-s + (−2.05 + 0.887i)10-s − 2.12i·11-s + (0.707 + 0.707i)12-s − 5.07·13-s + (1.84 + 1.84i)14-s + (2.07 + 0.823i)15-s + 16-s − 2.93i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.408 + 0.408i)3-s + 0.5·4-s + (0.917 − 0.397i)5-s + (−0.288 − 0.288i)6-s + (−0.697 − 0.697i)7-s − 0.353·8-s + 0.333i·9-s + (−0.648 + 0.280i)10-s − 0.639i·11-s + (0.204 + 0.204i)12-s − 1.40·13-s + (0.492 + 0.492i)14-s + (0.536 + 0.212i)15-s + 0.250·16-s − 0.711i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9479276835\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9479276835\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.05 + 0.887i)T \) |
| 37 | \( 1 + (2.66 + 5.46i)T \) |
good | 7 | \( 1 + (1.84 + 1.84i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.12iT - 11T^{2} \) |
| 13 | \( 1 + 5.07T + 13T^{2} \) |
| 17 | \( 1 + 2.93iT - 17T^{2} \) |
| 19 | \( 1 + (-2.28 + 2.28i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.76T + 23T^{2} \) |
| 29 | \( 1 + (5.76 + 5.76i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.38 + 1.38i)T - 31iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 2.00T + 43T^{2} \) |
| 47 | \( 1 + (-7.27 - 7.27i)T + 47iT^{2} \) |
| 53 | \( 1 + (0.286 - 0.286i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.25 - 8.25i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.44 + 2.44i)T - 61iT^{2} \) |
| 67 | \( 1 + (-9.89 + 9.89i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.69T + 71T^{2} \) |
| 73 | \( 1 + (5.92 + 5.92i)T + 73iT^{2} \) |
| 79 | \( 1 + (-5.73 + 5.73i)T - 79iT^{2} \) |
| 83 | \( 1 + (7.41 - 7.41i)T - 83iT^{2} \) |
| 89 | \( 1 + (-11.9 - 11.9i)T + 89iT^{2} \) |
| 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
−9.573720112987243795190023279582, −9.160358695779743707553763867010, −7.977565596065890224039014796827, −7.31378970668977104833156950630, −6.30372932718140813304244044812, −5.40815686605344455028581989909, −4.32539216536898971151701436195, −3.05386677126182535834057975586, −2.13448684970749194139582045975, −0.45718737311059567746623835997,
1.78834666083363936117602702589, 2.44435521635500627605710467853, 3.52610707455326835001055967147, 5.23362285759735013188081845319, 6.04509758889995246202078611766, 6.91701633327605467832781778457, 7.52175690493921399298849452631, 8.590745195304946277695559714830, 9.354444457973704191231118668477, 9.958110417800352027016596019530