| L(s) = 1 | + 2.30i·2-s + 1.30i·3-s − 3.30·4-s − 3·6-s + 4.30i·7-s − 3.00i·8-s + 1.30·9-s − 11-s − 4.30i·12-s − 5i·13-s − 9.90·14-s + 0.302·16-s − 3.90i·17-s + 3.00i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 1.62i·2-s + 0.752i·3-s − 1.65·4-s − 1.22·6-s + 1.62i·7-s − 1.06i·8-s + 0.434·9-s − 0.301·11-s − 1.24i·12-s − 1.38i·13-s − 2.64·14-s + 0.0756·16-s − 0.947i·17-s + 0.707i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.264232 - 1.11930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.264232 - 1.11930i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.30iT - 2T^{2} \) |
| 3 | \( 1 - 1.30iT - 3T^{2} \) |
| 7 | \( 1 - 4.30iT - 7T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 3.69iT - 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.60iT - 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 7.21iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 2.30iT - 53T^{2} \) |
| 59 | \( 1 + 0.211T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 2.90iT - 73T^{2} \) |
| 79 | \( 1 - 0.0916T + 79T^{2} \) |
| 83 | \( 1 + 14.5iT - 83T^{2} \) |
| 89 | \( 1 + 5.30T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−12.63238003338087815578628569761, −11.53923737131970850432248613390, −10.13341406256795812819042681010, −9.325478750161666109994915667218, −8.458271152466733221851348683016, −7.61899158499297047910126358421, −6.37652644928954917043497408833, −5.34985633576905330540675383112, −4.86749430750722988608237498817, −3.00248534429559080338269995500,
0.962669778807227584249628827249, 2.12667899427159159715711558370, 3.83197674752167377977467098631, 4.48061131817610434535676269095, 6.59836625902856050038281082045, 7.39720107290560321053135129321, 8.692065926056758252322454024284, 9.919407677304966816867197593895, 10.52701962670514038614560598389, 11.28844918856147401305385668142
