| L(s) = 1 | − 4i·2-s + (−10.8 − 11.2i)3-s − 16·4-s − 104.·5-s + (−44.9 + 43.2i)6-s + 64i·8-s + (−9.33 + 242. i)9-s + 419. i·10-s + 83.8i·11-s + (172. + 179. i)12-s − 382. i·13-s + (1.13e3 + 1.17e3i)15-s + 256·16-s − 1.43e3·17-s + (971. + 37.3i)18-s + 2.24e3i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.693 − 0.720i)3-s − 0.5·4-s − 1.87·5-s + (−0.509 + 0.490i)6-s + 0.353i·8-s + (−0.0384 + 0.999i)9-s + 1.32i·10-s + 0.208i·11-s + (0.346 + 0.360i)12-s − 0.628i·13-s + (1.29 + 1.35i)15-s + 0.250·16-s − 1.20·17-s + (0.706 + 0.0271i)18-s + 1.42i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.373 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3439256839\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3439256839\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4iT \) |
| 3 | \( 1 + (10.8 + 11.2i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 104.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 83.8iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 382. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.43e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.24e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.69e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.17e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.36e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.03e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.28e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.80e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 858. iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.19e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.90e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 3.70e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.39e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 5.24e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 4.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.44e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.65e4iT - 8.58e9T^{2} \) |
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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
−11.10624525611457227411542643411, −10.00628463283514185154544758066, −8.434018673047925058057673732121, −7.82443331517056686957039510968, −6.92355319503798119145526761319, −5.46249849096673015204109100532, −4.30359678062316363772355705700, −3.31738712515443015350632592539, −1.64345601480952341771490549808, −0.25856805584792329759853516258,
0.44720427278187012541325096198, 3.28127376128120027420217036433, 4.44571681934408645338207778002, 4.82545440493059264371289882214, 6.59678603115269440921519170358, 7.06287602753348041771750186843, 8.548228097463028054744212014163, 8.924620524802212093858793500480, 10.53367108472170535745678608379, 11.20545867458418565883333107380
