L(s) = 1 | + (0.292 − 1.70i)3-s + (−0.707 + 2.12i)5-s + (3 + 3i)7-s + (−2.82 − i)9-s + 4.24i·11-s + (−4 + 4i)13-s + (3.41 + 1.82i)15-s + (1.41 − 1.41i)17-s + (5.99 − 4.24i)21-s + (−3.99 − 3i)25-s + (−2.53 + 4.53i)27-s + 4.24·29-s + 6·31-s + (7.24 + 1.24i)33-s + (−8.48 + 4.24i)35-s + ⋯ |
L(s) = 1 | + (0.169 − 0.985i)3-s + (−0.316 + 0.948i)5-s + (1.13 + 1.13i)7-s + (−0.942 − 0.333i)9-s + 1.27i·11-s + (−1.10 + 1.10i)13-s + (0.881 + 0.472i)15-s + (0.342 − 0.342i)17-s + (1.30 − 0.925i)21-s + (−0.799 − 0.600i)25-s + (−0.487 + 0.872i)27-s + 0.787·29-s + 1.07·31-s + (1.26 + 0.216i)33-s + (−1.43 + 0.717i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26174 + 0.568997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26174 + 0.568997i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.292 + 1.70i)T \) |
| 5 | \( 1 + (0.707 - 2.12i)T \) |
good | 7 | \( 1 + (-3 - 3i)T + 7iT^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (4 - 4i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + (2 + 2i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-6 + 6i)T - 43iT^{2} \) |
| 47 | \( 1 + (-8.48 + 8.48i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.24T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 67iT^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + (5 - 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 8.48T + 89T^{2} \) |
| 97 | \( 1 + (5 + 5i)T + 97iT^{2} \) |
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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
−11.51627361908352457692614385487, −10.24982543651563735755543838411, −9.194661034841059892380859914713, −8.242082272601264509449990971401, −7.33604474520829536024053784969, −6.82602307122905347226975968649, −5.54748077000174391954610644579, −4.43851492764226680293398644322, −2.60840374490239961416477747319, −1.99120430608013858014842854958,
0.861039259749925024751000931242, 3.04685206407312178840235897360, 4.27498382748505261650323267559, 4.89574039209272713630942862647, 5.85114738184292384251303096517, 7.74944042557503591132607164393, 8.082857778880162557961481097085, 9.025921860170454464791271574191, 10.14635008178143324274567890495, 10.74527159248357242551966612633