L(s) = 1 | + 0.0802·7-s − 2.41i·11-s − 5.26i·13-s − 0.255·17-s − 6.95i·19-s + 1.64·23-s − 4.51i·29-s − 8.29·31-s − 2.67i·37-s + 8.11·41-s − 4.08i·43-s − 5.70·47-s − 6.99·49-s + 11.5i·53-s + 12.6i·59-s + ⋯ |
L(s) = 1 | + 0.0303·7-s − 0.728i·11-s − 1.46i·13-s − 0.0620·17-s − 1.59i·19-s + 0.343·23-s − 0.838i·29-s − 1.48·31-s − 0.439i·37-s + 1.26·41-s − 0.623i·43-s − 0.831·47-s − 0.999·49-s + 1.58i·53-s + 1.65i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8436360437\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8436360437\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.0802T + 7T^{2} \) |
| 11 | \( 1 + 2.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.26iT - 13T^{2} \) |
| 17 | \( 1 + 0.255T + 17T^{2} \) |
| 19 | \( 1 + 6.95iT - 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 4.51iT - 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 + 2.67iT - 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 + 4.08iT - 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 - 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 7.27iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 - 9.20iT - 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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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
−7.52383110235701490100392896168, −7.06076445417383116958970410992, −5.98743053482832328297485366590, −5.61847832289949751377012017894, −4.78739714353881420288083596203, −3.95539703098825704364960282027, −3.02852084018506243094877463724, −2.51794470497285361080117078930, −1.13443984574856060602858209684, −0.20851816740616069387154272685,
1.55128918278656377903905981653, 1.97600188551686354053529862991, 3.25253618768799978100663067731, 3.94344581023576470164595507664, 4.70003181107889705728655845493, 5.37760897208453247338478986601, 6.31301101425836518442191944204, 6.82195556094349576680258083941, 7.53840218117302506239167565799, 8.215268984625537204957048235277