| L(s) = 1 | + (7 − 12.1i)5-s + (2 + 3.46i)11-s + 54·13-s + (−7 − 12.1i)17-s + (−46 + 79.6i)19-s + (−76 + 131. i)23-s + (−35.5 − 61.4i)25-s + 106·29-s + (72 + 124. i)31-s + (−79 + 136. i)37-s + 390·41-s − 508·43-s + (−264 + 457. i)47-s + (303 + 524. i)53-s + 56·55-s + ⋯ |
| L(s) = 1 | + (0.626 − 1.08i)5-s + (0.0548 + 0.0949i)11-s + 1.15·13-s + (−0.0998 − 0.172i)17-s + (−0.555 + 0.962i)19-s + (−0.689 + 1.19i)23-s + (−0.284 − 0.491i)25-s + 0.678·29-s + (0.417 + 0.722i)31-s + (−0.351 + 0.607i)37-s + 1.48·41-s − 1.80·43-s + (−0.819 + 1.41i)47-s + (0.785 + 1.36i)53-s + 0.137·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.075661005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075661005\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-7 + 12.1i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 54T + 2.19e3T^{2} \) |
| 17 | \( 1 + (7 + 12.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (46 - 79.6i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (76 - 131. i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 106T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-72 - 124. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (79 - 136. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 390T + 6.89e4T^{2} \) |
| 43 | \( 1 + 508T + 7.95e4T^{2} \) |
| 47 | \( 1 + (264 - 457. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-303 - 524. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (182 + 315. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (339 - 587. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (422 + 730. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 8T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-211 - 365. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (192 - 332. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 548T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-597 + 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.50e3T + 9.12e5T^{2} \) |
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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
−9.000180299755715829088376851785, −8.390050338117211424068514403469, −7.66176092919759085609032099807, −6.39334461444201393752614273754, −5.89016383269228181608208425200, −4.98732680325032748249532463194, −4.17001431250774838681421756755, −3.14596983837836529175548715100, −1.71262069853415783459814326454, −1.14173216445116802085574670016,
0.44946874433453663507770217237, 1.91467595197289096487872979518, 2.71661798743668879415446923587, 3.69206658365482886912808801825, 4.66046360095585425227876365261, 5.85178418090096910570265142813, 6.46381411334078012928968298372, 6.95532128416389802725733227029, 8.187958782572275638078661158058, 8.716513940347302091798096182462
