| L(s) = 1 | + (−1.61 − 1.17i)2-s + (−1.33 + 4.09i)3-s + (1.23 + 3.80i)4-s + (6.52 − 4.73i)5-s + (6.97 − 5.06i)6-s + (8.05 + 24.8i)7-s + (2.47 − 7.60i)8-s + (6.82 + 4.95i)9-s − 16.1·10-s − 17.2·12-s + (−2.64 − 1.91i)13-s + (16.1 − 49.6i)14-s + (10.7 + 33.0i)15-s + (−12.9 + 9.40i)16-s + (−16.8 + 12.2i)17-s + (−5.21 − 16.0i)18-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.256 + 0.788i)3-s + (0.154 + 0.475i)4-s + (0.583 − 0.423i)5-s + (0.474 − 0.344i)6-s + (0.435 + 1.33i)7-s + (0.109 − 0.336i)8-s + (0.252 + 0.183i)9-s − 0.509·10-s − 0.414·12-s + (−0.0563 − 0.0409i)13-s + (0.307 − 0.946i)14-s + (0.184 + 0.568i)15-s + (−0.202 + 0.146i)16-s + (−0.240 + 0.175i)17-s + (−0.0682 − 0.210i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.970541 + 0.905298i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.970541 + 0.905298i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.61 + 1.17i)T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + (1.33 - 4.09i)T + (-21.8 - 15.8i)T^{2} \) |
| 5 | \( 1 + (-6.52 + 4.73i)T + (38.6 - 118. i)T^{2} \) |
| 7 | \( 1 + (-8.05 - 24.8i)T + (-277. + 201. i)T^{2} \) |
| 13 | \( 1 + (2.64 + 1.91i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (16.8 - 12.2i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-38.9 + 119. i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 - 97.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-81.5 - 250. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (161. + 117. i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-112. - 347. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (84.5 - 260. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 388.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (16.0 - 49.3i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-333. - 242. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-8.12 - 24.9i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-132. + 96.4i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 - 276.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-418. + 303. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-74.6 - 229. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (220. + 160. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-58.7 + 42.6i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.19e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.18e3 + 860. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.56748672218912711127682624837, −10.98581877184646551641834345256, −9.804820333025851103796833702686, −9.182280627798918611378811316374, −8.420706410173063327889764514634, −6.95253043673391322778329208786, −5.41472582863900529102604478184, −4.76893174015539985555313390781, −2.95441782250111880581647936573, −1.57399154039191841147324058917,
0.70558055547659924263823878325, 1.90598950028694218039775997549, 3.97940451019460234412269786095, 5.59338055178399292380537236114, 6.70869807314943789682242285006, 7.29608176980966985067200603442, 8.206290050005100412714193517852, 9.689946152328763568869137716716, 10.33092030505900234217095132163, 11.28136036071746395023682799114
