L(s) = 1 | + i·2-s + (−1.35 − 1.07i)3-s − 4-s + 0.745·5-s + (1.07 − 1.35i)6-s + (−2.35 − 1.20i)7-s − i·8-s + (0.697 + 2.91i)9-s + 0.745i·10-s + 2.89i·11-s + (1.35 + 1.07i)12-s − i·13-s + (1.20 − 2.35i)14-s + (−1.01 − 0.799i)15-s + 16-s + 2.53·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.785 − 0.619i)3-s − 0.5·4-s + 0.333·5-s + (0.438 − 0.555i)6-s + (−0.891 − 0.453i)7-s − 0.353i·8-s + (0.232 + 0.972i)9-s + 0.235i·10-s + 0.871i·11-s + (0.392 + 0.309i)12-s − 0.277i·13-s + (0.320 − 0.630i)14-s + (−0.261 − 0.206i)15-s + 0.250·16-s + 0.615·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.383169 + 0.598455i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.383169 + 0.598455i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.35 + 1.07i)T \) |
| 7 | \( 1 + (2.35 + 1.20i)T \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 - 0.745T + 5T^{2} \) |
| 11 | \( 1 - 2.89iT - 11T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 - 4.50iT - 19T^{2} \) |
| 23 | \( 1 - 5.62iT - 23T^{2} \) |
| 29 | \( 1 - 7.30iT - 29T^{2} \) |
| 31 | \( 1 - 6.18iT - 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 + 5.86T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 5.09iT - 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 9.20iT - 61T^{2} \) |
| 67 | \( 1 - 3.54T + 67T^{2} \) |
| 71 | \( 1 + 0.765iT - 71T^{2} \) |
| 73 | \( 1 - 6.83iT - 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + 1.62T + 89T^{2} \) |
| 97 | \( 1 + 1.89iT - 97T^{2} \) |
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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
−11.01167629714928872754078687770, −10.01317940574143049492733060796, −9.548510201930989682841965611405, −8.016054637592763031304942509790, −7.37083886307428907065574188619, −6.51002184508580456805760699994, −5.77287171677051432886076666126, −4.83468382893030827872522767693, −3.43274216038923850820835233689, −1.49412013703907662896748428617,
0.47373952575760258581957844043, 2.56815496984928208040744835403, 3.69346537843244631206472559670, 4.76906261724853992554117360009, 5.93408032469107112381203638219, 6.40979987083053404151386116938, 8.063238853630493717660333991240, 9.290652270098086934688934651316, 9.649596463315919453442376628826, 10.54149348302911815467277322152