| L(s) = 1 | + 2.77i·5-s − 2.56·7-s + 4.54i·11-s − 4.36i·13-s + 5.78i·17-s + (−1.44 − 4.11i)19-s + 6.51i·23-s − 2.69·25-s − 1.52·29-s − 8.55i·31-s − 7.10i·35-s + 10.9i·37-s − 2.73·41-s − 8.23·43-s − 0.591i·47-s + ⋯ |
| L(s) = 1 | + 1.24i·5-s − 0.967·7-s + 1.37i·11-s − 1.21i·13-s + 1.40i·17-s + (−0.331 − 0.943i)19-s + 1.35i·23-s − 0.538·25-s − 0.282·29-s − 1.53i·31-s − 1.20i·35-s + 1.80i·37-s − 0.427·41-s − 1.25·43-s − 0.0862i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2219689245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2219689245\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.44 + 4.11i)T \) |
| good | 5 | \( 1 - 2.77iT - 5T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 - 4.54iT - 11T^{2} \) |
| 13 | \( 1 + 4.36iT - 13T^{2} \) |
| 17 | \( 1 - 5.78iT - 17T^{2} \) |
| 23 | \( 1 - 6.51iT - 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 + 8.55iT - 31T^{2} \) |
| 37 | \( 1 - 10.9iT - 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 + 8.23T + 43T^{2} \) |
| 47 | \( 1 + 0.591iT - 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 13.2iT - 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + 2.57iT - 79T^{2} \) |
| 83 | \( 1 + 6.17iT - 83T^{2} \) |
| 89 | \( 1 - 8.90T + 89T^{2} \) |
| 97 | \( 1 + 0.565iT - 97T^{2} \) |
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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
−8.483557930376115121815880151066, −7.80630902500961518534333097941, −7.06552955952339119038110406894, −6.60060507557125559605535492831, −5.93520479473242608232149558565, −5.06469578488873916467081217021, −4.02064326092027872309545069569, −3.29848241430375639441425274296, −2.66709715194621876510680622902, −1.67647144442076374443729815161,
0.06430430653799741940266370485, 0.995024684896097788972447549183, 2.17745144575457150617658514834, 3.26621856598375674566511835592, 3.92647441653660939906607567759, 4.89251475661028394083878180513, 5.39171291522634918642047139057, 6.44748227997710372101084553142, 6.70659374714901641593669027769, 7.85887671729187450906575039229
