L(s) = 1 | + (−1.77 − 1.77i)3-s + (−2.23 + 0.115i)5-s + (−0.707 + 0.707i)7-s + 3.32i·9-s − 3.79i·11-s + (−3.76 + 3.76i)13-s + (4.17 + 3.76i)15-s + (4.22 + 4.22i)17-s + 2.26·19-s + 2.51·21-s + (−0.265 − 0.265i)23-s + (4.97 − 0.515i)25-s + (0.580 − 0.580i)27-s + 1.40i·29-s − 0.0693i·31-s + ⋯ |
L(s) = 1 | + (−1.02 − 1.02i)3-s + (−0.998 + 0.0516i)5-s + (−0.267 + 0.267i)7-s + 1.10i·9-s − 1.14i·11-s + (−1.04 + 1.04i)13-s + (1.07 + 0.972i)15-s + (1.02 + 1.02i)17-s + 0.519·19-s + 0.548·21-s + (−0.0553 − 0.0553i)23-s + (0.994 − 0.103i)25-s + (0.111 − 0.111i)27-s + 0.261i·29-s − 0.0124i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.439268 + 0.203331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.439268 + 0.203331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.23 - 0.115i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (1.77 + 1.77i)T + 3iT^{2} \) |
| 11 | \( 1 + 3.79iT - 11T^{2} \) |
| 13 | \( 1 + (3.76 - 3.76i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.22 - 4.22i)T + 17iT^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 + (0.265 + 0.265i)T + 23iT^{2} \) |
| 29 | \( 1 - 1.40iT - 29T^{2} \) |
| 31 | \( 1 + 0.0693iT - 31T^{2} \) |
| 37 | \( 1 + (-2.58 - 2.58i)T + 37iT^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 + (-8.35 - 8.35i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.39 - 6.39i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.32 + 5.32i)T - 53iT^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + (8.67 - 8.67i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.80iT - 71T^{2} \) |
| 73 | \( 1 + (7.40 - 7.40i)T - 73iT^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 + (9.62 + 9.62i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 + (7.58 + 7.58i)T + 97iT^{2} \) |
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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
−11.39261964275052884405336877675, −10.20961614809874187875362828730, −9.002826914752155667715413733502, −7.928930075232335259195674950876, −7.28911298592241456069739782117, −6.32854310508418229571874401970, −5.60838664214184359684616484491, −4.33029135242073370807625180286, −2.98892653683136659134377022535, −1.17330319409844315689373124951,
0.37087000487185379261609569769, 3.00512951000493187879191444538, 4.20832675620241438129990488544, 4.93061560405540032164672418198, 5.73242068788824812825602125767, 7.30141333948914684012297266748, 7.61300302333959867371708097513, 9.212575860782333265153286594159, 10.02440028506284950601564818492, 10.49279644718127719067891373935