| L(s) = 1 | + (2.23 − 2.23i)3-s + (2.23 + 2.23i)7-s − 7.00i·9-s + 10.0·21-s + (−6.70 + 6.70i)23-s + (−8.94 − 8.94i)27-s − 6i·29-s − 12·41-s + (−2.23 + 2.23i)43-s + (6.70 + 6.70i)47-s + 3.00i·49-s + 8·61-s + (15.6 − 15.6i)63-s + (11.1 + 11.1i)67-s + 30.0i·69-s + ⋯ |
| L(s) = 1 | + (1.29 − 1.29i)3-s + (0.845 + 0.845i)7-s − 2.33i·9-s + 2.18·21-s + (−1.39 + 1.39i)23-s + (−1.72 − 1.72i)27-s − 1.11i·29-s − 1.87·41-s + (−0.340 + 0.340i)43-s + (0.978 + 0.978i)47-s + 0.428i·49-s + 1.02·61-s + (1.97 − 1.97i)63-s + (1.36 + 1.36i)67-s + 3.61i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86810 - 1.04153i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86810 - 1.04153i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-2.23 + 2.23i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2.23 - 2.23i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (6.70 - 6.70i)T - 23iT^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + (2.23 - 2.23i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.70 - 6.70i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (-11.1 - 11.1i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-6.70 + 6.70i)T - 83iT^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−11.60827613628384846112671246941, −9.929877628441445663716608336962, −9.014301076137729022925882581596, −8.180222419607808861878002313188, −7.72231431178616882719109864936, −6.58946512478013834400918603139, −5.50338568631442837052196535536, −3.81878661239165248028630308085, −2.50440753921127709088745280024, −1.61449907431032737715089464787,
2.10597507111839493373218313936, 3.51443141489327993082822069688, 4.32505821517424656020472702238, 5.18623256982816384649673695548, 6.97309444263489073158465307755, 8.163335110556912520481640942854, 8.524893452988427596541792083614, 9.697404426810671569378454512657, 10.39988775557472225409398483677, 10.98357474495040166501479501267
