L(s) = 1 | + (1.30 + 0.533i)2-s + (−0.667 − 1.59i)3-s + (1.43 + 1.39i)4-s + (0.143 − 2.23i)5-s + (−0.0218 − 2.44i)6-s + (0.582 + 0.582i)7-s + (1.13 + 2.59i)8-s + (−2.10 + 2.13i)9-s + (1.37 − 2.84i)10-s − 3.68·11-s + (1.27 − 3.22i)12-s + (3.88 + 3.88i)13-s + (0.452 + 1.07i)14-s + (−3.66 + 1.26i)15-s + (0.0980 + 3.99i)16-s + (0.880 − 0.880i)17-s + ⋯ |
L(s) = 1 | + (0.926 + 0.377i)2-s + (−0.385 − 0.922i)3-s + (0.715 + 0.698i)4-s + (0.0639 − 0.997i)5-s + (−0.00892 − 0.999i)6-s + (0.220 + 0.220i)7-s + (0.399 + 0.916i)8-s + (−0.703 + 0.711i)9-s + (0.435 − 0.900i)10-s − 1.11·11-s + (0.368 − 0.929i)12-s + (1.07 + 1.07i)13-s + (0.120 + 0.287i)14-s + (−0.945 + 0.325i)15-s + (0.0245 + 0.999i)16-s + (0.213 − 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51533 - 0.209406i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51533 - 0.209406i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.30 - 0.533i)T \) |
| 3 | \( 1 + (0.667 + 1.59i)T \) |
| 5 | \( 1 + (-0.143 + 2.23i)T \) |
good | 7 | \( 1 + (-0.582 - 0.582i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + (-3.88 - 3.88i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.880 + 0.880i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 + (2.06 + 2.06i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.37iT - 29T^{2} \) |
| 31 | \( 1 - 3.32T + 31T^{2} \) |
| 37 | \( 1 + (-2.44 + 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.648iT - 41T^{2} \) |
| 43 | \( 1 + (-0.819 - 0.819i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.28 + 6.28i)T - 47iT^{2} \) |
| 53 | \( 1 + (5.60 - 5.60i)T - 53iT^{2} \) |
| 59 | \( 1 - 6.12iT - 59T^{2} \) |
| 61 | \( 1 + 5.13iT - 61T^{2} \) |
| 67 | \( 1 + (-4.90 + 4.90i)T - 67iT^{2} \) |
| 71 | \( 1 - 4.13iT - 71T^{2} \) |
| 73 | \( 1 + (-4.69 + 4.69i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.10iT - 79T^{2} \) |
| 83 | \( 1 + (-6.27 + 6.27i)T - 83iT^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + (5.42 + 5.42i)T + 97iT^{2} \) |
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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
−13.33831958070264086598373339496, −12.61734743026030683672195432286, −11.78639301127205022810939649004, −10.78115910714502190552748340618, −8.658372123498018801022361761392, −7.934453557309450302437162928293, −6.52111757687582266394653844896, −5.57711097469134798338048569281, −4.39260441000675668994751170338, −2.12504066730631169563215257345,
2.87766528510241903561645032195, 4.06399524049924882786686917120, 5.50495952845328401653251652457, 6.38474815530948889890063050272, 8.041914901953000438514649035390, 9.949487679612763205150627666707, 10.74135761758300997532925488283, 11.08916711967605539892531703035, 12.52681707905928159612431731780, 13.57657404869243726835871235926