| L(s) = 1 | − 1.56·3-s + (2 + i)5-s + (2 − 1.73i)7-s − 0.561·9-s − 6.16·11-s + 3.56i·13-s + (−3.12 − 1.56i)15-s − 2.70·17-s − 7.12i·19-s + (−3.12 + 2.70i)21-s + 7.12·23-s + (3 + 4i)25-s + 5.56·27-s + 2.70i·29-s + 5.40·31-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + (0.894 + 0.447i)5-s + (0.755 − 0.654i)7-s − 0.187·9-s − 1.85·11-s + 0.987i·13-s + (−0.806 − 0.403i)15-s − 0.655·17-s − 1.63i·19-s + (−0.681 + 0.590i)21-s + 1.48·23-s + (0.600 + 0.800i)25-s + 1.07·27-s + 0.502i·29-s + 0.971·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.350386116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.350386116\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 6.16T + 11T^{2} \) |
| 13 | \( 1 - 3.56iT - 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 7.12iT - 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 2.70iT - 29T^{2} \) |
| 31 | \( 1 - 5.40T + 31T^{2} \) |
| 37 | \( 1 - 5.40T + 37T^{2} \) |
| 41 | \( 1 + 5.40iT - 41T^{2} \) |
| 43 | \( 1 - 12.3iT - 43T^{2} \) |
| 47 | \( 1 + 6.16iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 2.24iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 4.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 5.40iT - 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−9.048756079268859483950444653753, −8.298120523294744573426812739120, −7.16433644991436868168941571050, −6.79475601371255508242759882100, −5.83115399771717892287593167385, −4.85464585978489879950823002134, −4.79642292789501751450558614691, −2.95911383502731901304474202684, −2.23983836926578256591645483181, −0.76327084268047406487575906868,
0.800422804074571276252252057458, 2.19718740430590899509636636532, 2.94062844877391035103428907151, 4.68749158110065515525246310506, 5.23649315494443252420481609884, 5.70177114465155528759639658420, 6.34007455433652175972693957449, 7.69438594228907210628610001692, 8.240312772650704606256502283689, 8.932459661056164285197314641205
