| L(s) = 1 | + 2-s + 4-s + (0.526 − 2.17i)5-s + 8-s + (0.526 − 2.17i)10-s + 4.41i·11-s − 4.94·13-s + 16-s − 5.22i·17-s + 4.97i·19-s + (0.526 − 2.17i)20-s + 4.41i·22-s − 7.71·23-s + (−4.44 − 2.28i)25-s − 4.94·26-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.235 − 0.971i)5-s + 0.353·8-s + (0.166 − 0.687i)10-s + 1.33i·11-s − 1.37·13-s + 0.250·16-s − 1.26i·17-s + 1.14i·19-s + (0.117 − 0.485i)20-s + 0.942i·22-s − 1.60·23-s + (−0.889 − 0.457i)25-s − 0.969·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.544844325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.544844325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.526 + 2.17i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4.41iT - 11T^{2} \) |
| 13 | \( 1 + 4.94T + 13T^{2} \) |
| 17 | \( 1 + 5.22iT - 17T^{2} \) |
| 19 | \( 1 - 4.97iT - 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 - 6.43iT - 31T^{2} \) |
| 37 | \( 1 - 9.25iT - 37T^{2} \) |
| 41 | \( 1 + 6.79T + 41T^{2} \) |
| 43 | \( 1 - 2.08iT - 43T^{2} \) |
| 47 | \( 1 - 7.83iT - 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 6.76iT - 61T^{2} \) |
| 67 | \( 1 + 2.40iT - 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 + 0.202T + 73T^{2} \) |
| 79 | \( 1 + 0.736T + 79T^{2} \) |
| 83 | \( 1 + 3.96iT - 83T^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−8.471659077982665929475711518320, −7.70623414657804653809392807965, −7.09499936634888024293490749133, −6.34876195930956645657332745236, −5.17475298337687525336656463452, −5.02864208357451028663335606064, −4.28331695764140178421592473556, −3.23762706776460397811517756369, −2.18081854009189072357093401485, −1.45993261696412088548165014292,
0.29621472786686181828003676346, 2.21656564257801525966386263715, 2.49883353975554690542429657607, 3.74261623067521212025294855332, 4.10807671364645874807918844573, 5.43557408350462276152590509488, 5.87038473525497051238209615702, 6.55535153231226475543383244376, 7.31616674169989229609827209052, 7.985646709656977944941931340106
