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
−10.10217442216252437208880898820, −9.582094635553581780732232920337, −8.544008996603335232484989430792, −8.259341364298413972612336163706, −7.11243114117088627066795860684, −6.05981964190008158402349586449, −5.36016801541055784270403078110, −4.10211779810371331364174026312, −3.53799915056726608342183563231, −1.88484035872066183219589543785,
0.18923494939681368925339628273, 1.63444556036362915857820698542, 2.91596499658786261719862556959, 3.47369422736389106747303190811, 4.98643908074393308513286618613, 6.33363952668499342447079985573, 6.99729265678542535990371490827, 7.65984763750919418448846136304, 8.607759375025249282399941381437, 9.245257464606355390337547264419