L(s) = 1 | + (−2.17 + 3.77i)2-s + (−5.5 − 9.52i)4-s + (4.35 − 7.54i)5-s + 13.0·8-s + (19 + 32.9i)10-s + (21.7 + 37.7i)11-s − 82·13-s + (15.4 − 26.8i)16-s + (−39.2 − 67.9i)17-s + (−10 + 17.3i)19-s − 95.8·20-s − 190·22-s + (65.3 − 113. i)23-s + (24.5 + 42.4i)25-s + (178. − 309. i)26-s + ⋯ |
L(s) = 1 | + (−0.770 + 1.33i)2-s + (−0.687 − 1.19i)4-s + (0.389 − 0.675i)5-s + 0.577·8-s + (0.600 + 1.04i)10-s + (0.597 + 1.03i)11-s − 1.74·13-s + (0.242 − 0.419i)16-s + (−0.559 − 0.969i)17-s + (−0.120 + 0.209i)19-s − 1.07·20-s − 1.84·22-s + (0.592 − 1.02i)23-s + (0.196 + 0.339i)25-s + (1.34 − 2.33i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.122298217\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.122298217\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (2.17 - 3.77i)T + (-4 - 6.92i)T^{2} \) |
| 5 | \( 1 + (-4.35 + 7.54i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-21.7 - 37.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 82T + 2.19e3T^{2} \) |
| 17 | \( 1 + (39.2 + 67.9i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (10 - 17.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-65.3 + 113. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 244.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-78 - 135. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (93 - 161. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 165.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-235. + 407. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-78.4 - 135. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-78.4 - 135. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-395 + 684. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-22 - 38.1i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 444.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-63 - 109. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-356 + 616. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (727. - 1.26e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 798T + 9.12e5T^{2} \) |
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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
−10.39997970648041278276382764684, −9.572593750411892294952441579873, −9.071332091230382076535127629606, −8.130292303832499312502980318786, −7.03012493834074643204676113849, −6.67779846873393102312484788003, −5.14620485606613983676676481588, −4.71061173254028284909299407118, −2.48219143815817763611386772638, −0.74632344157517316768780159897,
0.76063709256205766346453270210, 2.23409348892514736247558520121, 2.99135492943705428290394441905, 4.25526337361325856547525963808, 5.84408234879797842783003256581, 6.88434421564925110482535747244, 8.109878075986695048430055007137, 9.054164961378004778225640947165, 9.775444060253388423050829933524, 10.56849470925823827235536611044