| L(s) = 1 | + 0.150i·2-s + 1.96i·3-s + 1.97·4-s − 0.296·6-s − 1.54i·7-s + 0.600i·8-s − 0.849·9-s + 4.56·11-s + 3.87i·12-s + 1.09i·13-s + 0.233·14-s + 3.86·16-s + i·17-s − 0.128i·18-s − 4.67·19-s + ⋯ |
| L(s) = 1 | + 0.106i·2-s + 1.13i·3-s + 0.988·4-s − 0.120·6-s − 0.583i·7-s + 0.212i·8-s − 0.283·9-s + 1.37·11-s + 1.11i·12-s + 0.304i·13-s + 0.0623·14-s + 0.965·16-s + 0.242i·17-s − 0.0302i·18-s − 1.07·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54408 + 0.954295i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54408 + 0.954295i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 2 | \( 1 - 0.150iT - 2T^{2} \) |
| 3 | \( 1 - 1.96iT - 3T^{2} \) |
| 7 | \( 1 + 1.54iT - 7T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 13 | \( 1 - 1.09iT - 13T^{2} \) |
| 19 | \( 1 + 4.67T + 19T^{2} \) |
| 23 | \( 1 + 0.529iT - 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 + 4.78T + 31T^{2} \) |
| 37 | \( 1 + 5.27iT - 37T^{2} \) |
| 41 | \( 1 + 0.751T + 41T^{2} \) |
| 43 | \( 1 + 9.49iT - 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + 0.0227iT - 53T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 - 3.92T + 61T^{2} \) |
| 67 | \( 1 + 9.75iT - 67T^{2} \) |
| 71 | \( 1 - 1.21T + 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 5.08iT - 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 6.78iT - 97T^{2} \) |
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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
−10.99371893205491112033154572053, −10.66639441673801387158443765402, −9.580891788732424596130398534834, −8.850075796976703700708405843538, −7.47140643007903925370836702414, −6.68587841098857547942570681915, −5.65732963866203563206532723727, −4.22766031583635413407810924574, −3.61385148795421309198937936320, −1.82903636398634276022821237747,
1.42787677211705478308065278761, 2.39025814567206412522455308354, 3.84224518791347240589108515497, 5.64712074246121089214438731636, 6.52965739628397926522451327921, 7.10431974426411427134195336480, 8.089952988190842236149765567328, 9.107841460276900357607707390719, 10.22837906881227425367272097961, 11.42125172928357514610827934153
