L(s) = 1 | − 2·11-s − 13-s + 2·17-s − 5·19-s − 6·23-s − 5·25-s − 8·29-s − 3·31-s + 9·37-s − 2·41-s − 43-s − 8·47-s + 6·53-s + 6·59-s + 2·61-s + 5·67-s − 4·71-s − 11·73-s − 5·79-s − 12·89-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 0.603·11-s − 0.277·13-s + 0.485·17-s − 1.14·19-s − 1.25·23-s − 25-s − 1.48·29-s − 0.538·31-s + 1.47·37-s − 0.312·41-s − 0.152·43-s − 1.16·47-s + 0.824·53-s + 0.781·59-s + 0.256·61-s + 0.610·67-s − 0.474·71-s − 1.28·73-s − 0.562·79-s − 1.27·89-s + 1.82·97-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
\(L(1)\) |
\(\approx\) |
\(0.8958526736\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8958526736\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−15.12323877891526, −14.68510119913930, −14.31543890422901, −13.45992436348571, −13.08390371935772, −12.70936912350252, −11.87862958364626, −11.57166769148048, −10.88211616826204, −10.33475564459001, −9.752347212119945, −9.420092835482481, −8.461240801675830, −8.133856578282871, −7.517608444549793, −6.986449964715824, −6.103656658764243, −5.751870550276044, −5.113990881780336, −4.207842894320326, −3.887212452883653, −2.966191144976456, −2.189334791314179, −1.665964223362302, −0.3480322191045424,
0.3480322191045424, 1.665964223362302, 2.189334791314179, 2.966191144976456, 3.887212452883653, 4.207842894320326, 5.113990881780336, 5.751870550276044, 6.103656658764243, 6.986449964715824, 7.517608444549793, 8.133856578282871, 8.461240801675830, 9.420092835482481, 9.752347212119945, 10.33475564459001, 10.88211616826204, 11.57166769148048, 11.87862958364626, 12.70936912350252, 13.08390371935772, 13.45992436348571, 14.31543890422901, 14.68510119913930, 15.12323877891526