| L(s) = 1 | + (2.66 − 3.67i)3-s + (−0.873 + 11.1i)5-s + 36.0i·7-s + (1.97 + 6.08i)9-s + (9.65 − 29.7i)11-s + (72.2 − 23.4i)13-s + (38.6 + 32.9i)15-s + (−8.51 − 11.7i)17-s + (−41.2 + 29.9i)19-s + (132. + 96.3i)21-s + (−113. − 37.0i)23-s + (−123. − 19.4i)25-s + (144. + 46.8i)27-s + (167. + 121. i)29-s + (−19.4 + 14.1i)31-s + ⋯ |
| L(s) = 1 | + (0.513 − 0.706i)3-s + (−0.0780 + 0.996i)5-s + 1.94i·7-s + (0.0732 + 0.225i)9-s + (0.264 − 0.814i)11-s + (1.54 − 0.500i)13-s + (0.664 + 0.567i)15-s + (−0.121 − 0.167i)17-s + (−0.498 + 0.361i)19-s + (1.37 + 1.00i)21-s + (−1.03 − 0.335i)23-s + (−0.987 − 0.155i)25-s + (1.02 + 0.333i)27-s + (1.07 + 0.777i)29-s + (−0.112 + 0.0818i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.70701 + 0.645759i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70701 + 0.645759i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.873 - 11.1i)T \) |
| good | 3 | \( 1 + (-2.66 + 3.67i)T + (-8.34 - 25.6i)T^{2} \) |
| 7 | \( 1 - 36.0iT - 343T^{2} \) |
| 11 | \( 1 + (-9.65 + 29.7i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-72.2 + 23.4i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (8.51 + 11.7i)T + (-1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (41.2 - 29.9i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (113. + 37.0i)T + (9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-167. - 121. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (19.4 - 14.1i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (82.8 - 26.9i)T + (4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-57.4 - 176. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 243. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-318. + 437. i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-289. + 398. i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (171. + 529. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (97.4 - 299. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (59.2 + 81.6i)T + (-9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (27.9 + 20.2i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-996. - 323. i)T + (3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (265. + 192. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (559. + 769. i)T + (-1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-184. + 568. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-469. + 645. i)T + (-2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.60732656315397535721121323719, −12.47600442774865163216866675886, −11.49948307036827101119424899458, −10.42192981195435686706441194817, −8.691751867340127571578321546182, −8.246560017497897693887319621469, −6.58226101672113144458156295983, −5.71351213023529715585407483251, −3.27619996771546025591835414951, −2.09699599550203851789555235215,
1.12241413070809759539337253828, 4.02118496527681734106926439145, 4.27816819497762018881342708187, 6.46564543961404519184123794262, 7.84930374253310581998124786673, 9.002899191597524899035616110447, 9.950144440478687237399325734827, 10.91194513725783020407931708225, 12.32401539251382789138930475111, 13.52508107516279358116895924801
