L(s) = 1 | − 5-s + 2·7-s − 3·9-s + 6·11-s + 2·13-s − 2·17-s + 2·19-s − 2·23-s + 25-s − 29-s − 2·31-s − 2·35-s + 10·37-s + 2·41-s − 8·43-s + 3·45-s + 12·47-s − 3·49-s − 6·53-s − 6·55-s + 8·59-s − 6·61-s − 6·63-s − 2·65-s − 2·67-s + 12·71-s − 6·73-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.755·7-s − 9-s + 1.80·11-s + 0.554·13-s − 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s − 0.185·29-s − 0.359·31-s − 0.338·35-s + 1.64·37-s + 0.312·41-s − 1.21·43-s + 0.447·45-s + 1.75·47-s − 3/7·49-s − 0.824·53-s − 0.809·55-s + 1.04·59-s − 0.768·61-s − 0.755·63-s − 0.248·65-s − 0.244·67-s + 1.42·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.872289595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.872289595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.998855934638511304543288379208, −8.257954938132273132829266439629, −7.59896035728415497254929554682, −6.55138204331491112705378355339, −6.00486449332409074941837987010, −4.95898829319521499302586493249, −4.08126465874364407576578580037, −3.38444605735957405294650498855, −2.08605131316588777212252155975, −0.921963524041875158976208258286,
0.921963524041875158976208258286, 2.08605131316588777212252155975, 3.38444605735957405294650498855, 4.08126465874364407576578580037, 4.95898829319521499302586493249, 6.00486449332409074941837987010, 6.55138204331491112705378355339, 7.59896035728415497254929554682, 8.257954938132273132829266439629, 8.998855934638511304543288379208