| L(s) = 1 | + (1 − 1.73i)2-s + (−1.5 − 2.59i)3-s + (−1.99 − 3.46i)4-s + (−4 + 6.92i)5-s − 6·6-s − 7.99·8-s + (−4.5 + 7.79i)9-s + (7.99 + 13.8i)10-s + (−20 − 34.6i)11-s + (−6.00 + 10.3i)12-s + 4·13-s + 24·15-s + (−8 + 13.8i)16-s + (42 + 72.7i)17-s + (9 + 15.5i)18-s + (−74 + 128. i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.357 + 0.619i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.252 + 0.438i)10-s + (−0.548 − 0.949i)11-s + (−0.144 + 0.249i)12-s + 0.0853·13-s + 0.413·15-s + (−0.125 + 0.216i)16-s + (0.599 + 1.03i)17-s + (0.117 + 0.204i)18-s + (−0.893 + 1.54i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9480705900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9480705900\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 + (1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (4 - 6.92i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (20 + 34.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-42 - 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (74 - 128. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (42 - 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 58T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-68 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-111 + 192. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 420T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (244 - 422. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (239 + 413. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (274 + 474. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (346 - 599. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-454 - 786. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 524T + 3.57e5T^{2} \) |
| 73 | \( 1 + (220 + 381. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (608 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 684T + 5.71e5T^{2} \) |
| 89 | \( 1 + (302 - 523. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 832T + 9.12e5T^{2} \) |
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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
−11.38543247386494693599503565393, −10.79034411428027791958373719113, −9.976374267976122901455145382724, −8.459118737224851503611659486033, −7.69848524372415640276775106786, −6.29313344313762492443785812238, −5.61003788151056195473730988689, −3.99815560761166531571864980821, −2.96603995273747560813670634733, −1.43973712575338838613327973047,
0.33752015792507152696621433121, 2.75456330137977340038610525385, 4.52140171966288868192363217261, 4.77684924651452475226130786052, 6.17062570241153316956728462472, 7.25643954537215780077776624808, 8.267948789633436406310289145424, 9.231594882267329263876464981298, 10.17327205337815577873249750054, 11.33668813506280532538537503202
