| L(s) = 1 | + (0.445 − 1.94i)2-s + (4.72 − 5.92i)3-s + (−3.60 − 1.73i)4-s + (8.54 − 10.7i)5-s + (−9.45 − 11.8i)6-s + (13.8 + 12.2i)7-s + (−4.98 + 6.25i)8-s + (−6.78 − 29.7i)9-s + (−17.0 − 21.4i)10-s + (−0.842 + 3.69i)11-s + (−27.3 + 13.1i)12-s + (−10.5 + 46.4i)13-s + (30.1 − 21.5i)14-s + (−23.1 − 101. i)15-s + (9.97 + 12.5i)16-s + (−100. + 48.2i)17-s + ⋯ |
| L(s) = 1 | + (0.157 − 0.689i)2-s + (0.909 − 1.14i)3-s + (−0.450 − 0.216i)4-s + (0.764 − 0.958i)5-s + (−0.643 − 0.806i)6-s + (0.748 + 0.663i)7-s + (−0.220 + 0.276i)8-s + (−0.251 − 1.10i)9-s + (−0.540 − 0.677i)10-s + (−0.0230 + 0.101i)11-s + (−0.657 + 0.316i)12-s + (−0.226 + 0.990i)13-s + (0.574 − 0.411i)14-s + (−0.398 − 1.74i)15-s + (0.155 + 0.195i)16-s + (−1.42 + 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20553 - 2.07986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20553 - 2.07986i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.445 + 1.94i)T \) |
| 7 | \( 1 + (-13.8 - 12.2i)T \) |
| good | 3 | \( 1 + (-4.72 + 5.92i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-8.54 + 10.7i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (0.842 - 3.69i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (10.5 - 46.4i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (100. - 48.2i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (141. + 68.2i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (19.9 - 9.62i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 93.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-302. + 145. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-158. + 198. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-109. - 137. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (127. - 556. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-71.6 - 34.4i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (417. + 523. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-260. + 125. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 191.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (366. + 176. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (146. + 640. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 852.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (219. + 960. i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (71.6 + 313. i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−12.99680062192318756844768207884, −12.35467311449670716808636868667, −11.25392060101394750186111831884, −9.421705228723640048838374821942, −8.839180032672687018837332881457, −7.73595792045920422504035128459, −6.03754767477267881514151879608, −4.54800049209531699639866727900, −2.32187445991300097199075933469, −1.53882795123501433862438168153,
2.79091967563018772987964144813, 4.17852390499142675968463149274, 5.51614721978153959158460771823, 7.13534167251948466605865647342, 8.217557878194117814128340694019, 9.553213177209559465077323280565, 10.21256976563693001214674916815, 11.35486677644786589236832451632, 13.48502665995721816689018862195, 13.98404850076743975415023771203
