| L(s) = 1 | − 5-s − 11-s + 4·13-s + 2·17-s − 6·19-s − 5·23-s − 4·25-s − 10·29-s + 31-s − 5·37-s − 2·41-s + 8·43-s − 8·47-s + 6·53-s + 55-s − 3·59-s + 2·61-s − 4·65-s + 3·67-s + 71-s − 10·73-s − 6·79-s − 12·83-s − 2·85-s − 15·89-s + 6·95-s + 5·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s − 1.04·23-s − 4/5·25-s − 1.85·29-s + 0.179·31-s − 0.821·37-s − 0.312·41-s + 1.21·43-s − 1.16·47-s + 0.824·53-s + 0.134·55-s − 0.390·59-s + 0.256·61-s − 0.496·65-s + 0.366·67-s + 0.118·71-s − 1.17·73-s − 0.675·79-s − 1.31·83-s − 0.216·85-s − 1.58·89-s + 0.615·95-s + 0.507·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6577773766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6577773766\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.06615716047900, −13.46578962908126, −13.02528223314535, −12.59945932130193, −12.03436267153365, −11.44266017539085, −11.11896774392259, −10.56275990110637, −10.05154900261223, −9.543559372733081, −8.840502180585382, −8.399843911338673, −8.002184899850353, −7.411768873609313, −6.885019958431739, −6.139301730239463, −5.784867156918083, −5.275045873018815, −4.278992076577747, −4.017555076051791, −3.500692225967257, −2.707951088006415, −1.918200282508229, −1.431781456881217, −0.2585259273034474,
0.2585259273034474, 1.431781456881217, 1.918200282508229, 2.707951088006415, 3.500692225967257, 4.017555076051791, 4.278992076577747, 5.275045873018815, 5.784867156918083, 6.139301730239463, 6.885019958431739, 7.411768873609313, 8.002184899850353, 8.399843911338673, 8.840502180585382, 9.543559372733081, 10.05154900261223, 10.56275990110637, 11.11896774392259, 11.44266017539085, 12.03436267153365, 12.59945932130193, 13.02528223314535, 13.46578962908126, 14.06615716047900
