L(s) = 1 | − 5-s + 1.64·11-s − 2.64·13-s + 1.64·17-s + 8.29·19-s − 1.64·23-s + 25-s − 7.64·29-s − 4.29·31-s + 0.645·37-s − 4.93·41-s − 5.93·43-s − 6·47-s − 3.29·53-s − 1.64·55-s − 10.9·59-s + 8·61-s + 2.64·65-s + 0.645·67-s − 13.6·71-s + 13.2·73-s + 2.29·79-s + 10.9·83-s − 1.64·85-s + 14.2·89-s − 8.29·95-s + 8·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.496·11-s − 0.733·13-s + 0.399·17-s + 1.90·19-s − 0.343·23-s + 0.200·25-s − 1.41·29-s − 0.770·31-s + 0.106·37-s − 0.771·41-s − 0.905·43-s − 0.875·47-s − 0.452·53-s − 0.221·55-s − 1.42·59-s + 1.02·61-s + 0.328·65-s + 0.0788·67-s − 1.61·71-s + 1.54·73-s + 0.257·79-s + 1.20·83-s − 0.178·85-s + 1.50·89-s − 0.850·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 - 1.64T + 17T^{2} \) |
| 19 | \( 1 - 8.29T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 + 7.64T + 29T^{2} \) |
| 31 | \( 1 + 4.29T + 31T^{2} \) |
| 37 | \( 1 - 0.645T + 37T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 0.645T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 2.29T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−7.56596615220735293610191394512, −6.84918876265612676064067814356, −6.02610987945194812004130463538, −5.21958713038120697220576614402, −4.76415803825304750978927478678, −3.56359022827858222592413616982, −3.38262089798596278931637216823, −2.14762124679489739752420359813, −1.23313944687777422143244160072, 0,
1.23313944687777422143244160072, 2.14762124679489739752420359813, 3.38262089798596278931637216823, 3.56359022827858222592413616982, 4.76415803825304750978927478678, 5.21958713038120697220576614402, 6.02610987945194812004130463538, 6.84918876265612676064067814356, 7.56596615220735293610191394512