L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.544 − 1.64i)3-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1.54 + 0.778i)6-s − 1.56·7-s + (0.707 − 0.707i)8-s + (−2.40 − 1.78i)9-s + 1.00·10-s + 1.64·11-s + (1.64 + 0.544i)12-s + (−0.629 − 0.629i)13-s + (1.10 + 1.10i)14-s + (0.778 + 1.54i)15-s − 1.00·16-s + (4.46 − 4.46i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.314 − 0.949i)3-s + 0.500i·4-s + (−0.316 + 0.316i)5-s + (−0.631 + 0.317i)6-s − 0.592·7-s + (0.250 − 0.250i)8-s + (−0.802 − 0.596i)9-s + 0.316·10-s + 0.495·11-s + (0.474 + 0.157i)12-s + (−0.174 − 0.174i)13-s + (0.296 + 0.296i)14-s + (0.200 + 0.399i)15-s − 0.250·16-s + (1.08 − 1.08i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3899956936\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3899956936\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.544 + 1.64i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 37 | \( 1 + (5.36 - 2.85i)T \) |
good | 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 + (0.629 + 0.629i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.46 + 4.46i)T - 17iT^{2} \) |
| 19 | \( 1 + (4.49 + 4.49i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.08 - 1.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (-0.289 - 0.289i)T + 29iT^{2} \) |
| 31 | \( 1 + (5.62 - 5.62i)T - 31iT^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + (0.834 + 0.834i)T + 43iT^{2} \) |
| 47 | \( 1 - 13.1iT - 47T^{2} \) |
| 53 | \( 1 + 4.14iT - 53T^{2} \) |
| 59 | \( 1 + (-8.55 + 8.55i)T - 59iT^{2} \) |
| 61 | \( 1 + (9.70 - 9.70i)T - 61iT^{2} \) |
| 67 | \( 1 + 4.75iT - 67T^{2} \) |
| 71 | \( 1 + 3.82iT - 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 + (0.689 + 0.689i)T + 79iT^{2} \) |
| 83 | \( 1 + 2.00iT - 83T^{2} \) |
| 89 | \( 1 + (7.48 + 7.48i)T + 89iT^{2} \) |
| 97 | \( 1 + (-3.88 - 3.88i)T + 97iT^{2} \) |
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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
−9.245257464606355390337547264419, −8.607759375025249282399941381437, −7.65984763750919418448846136304, −6.99729265678542535990371490827, −6.33363952668499342447079985573, −4.98643908074393308513286618613, −3.47369422736389106747303190811, −2.91596499658786261719862556959, −1.63444556036362915857820698542, −0.18923494939681368925339628273,
1.88484035872066183219589543785, 3.53799915056726608342183563231, 4.10211779810371331364174026312, 5.36016801541055784270403078110, 6.05981964190008158402349586449, 7.11243114117088627066795860684, 8.259341364298413972612336163706, 8.544008996603335232484989430792, 9.582094635553581780732232920337, 10.10217442216252437208880898820