| L(s) = 1 | + 4·5-s − 2·17-s + 2·19-s + 8·23-s + 11·25-s + 2·29-s + 4·31-s + 6·37-s − 2·41-s + 8·43-s + 4·47-s − 10·53-s + 6·59-s + 4·61-s − 12·67-s + 14·73-s + 8·79-s + 6·83-s − 8·85-s + 10·89-s + 8·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.485·17-s + 0.458·19-s + 1.66·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s + 0.986·37-s − 0.312·41-s + 1.21·43-s + 0.583·47-s − 1.37·53-s + 0.781·59-s + 0.512·61-s − 1.46·67-s + 1.63·73-s + 0.900·79-s + 0.658·83-s − 0.867·85-s + 1.05·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.163983536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.163983536\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.04210138266322, −14.68577375341962, −13.97289100955479, −13.68487021499409, −13.14062563310171, −12.75034310294143, −12.10709190110319, −11.33969513882305, −10.74896746303824, −10.43184171837922, −9.590775330081840, −9.378872929220017, −8.875722562175143, −8.141100523200548, −7.389534426986802, −6.678531276646575, −6.324649608176539, −5.682942734983359, −5.055195372199618, −4.653457671635681, −3.625181531107512, −2.680435722611640, −2.452465782425863, −1.432097211330738, −0.8563809403152163,
0.8563809403152163, 1.432097211330738, 2.452465782425863, 2.680435722611640, 3.625181531107512, 4.653457671635681, 5.055195372199618, 5.682942734983359, 6.324649608176539, 6.678531276646575, 7.389534426986802, 8.141100523200548, 8.875722562175143, 9.378872929220017, 9.590775330081840, 10.43184171837922, 10.74896746303824, 11.33969513882305, 12.10709190110319, 12.75034310294143, 13.14062563310171, 13.68487021499409, 13.97289100955479, 14.68577375341962, 15.04210138266322
