L(s) = 1 | − 2·5-s − 2·7-s − 3·9-s − 2·11-s + 6·17-s − 6·19-s − 8·23-s − 25-s + 2·29-s + 10·31-s + 4·35-s + 6·37-s + 6·41-s − 4·43-s + 6·45-s − 2·47-s − 3·49-s + 6·53-s + 4·55-s − 10·59-s − 2·61-s + 6·63-s + 10·67-s + 10·71-s − 2·73-s + 4·77-s + 4·79-s + ⋯ |
L(s) = 1 | − 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s + 1.45·17-s − 1.37·19-s − 1.66·23-s − 1/5·25-s + 0.371·29-s + 1.79·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 0.894·45-s − 0.291·47-s − 3/7·49-s + 0.824·53-s + 0.539·55-s − 1.30·59-s − 0.256·61-s + 0.755·63-s + 1.22·67-s + 1.18·71-s − 0.234·73-s + 0.455·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7772222947\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7772222947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.564530664989272348404745281383, −8.091725484064501883319978493826, −7.57960106576853046343584865094, −6.27574617805035798262057908141, −6.02684210173332234104860151617, −4.86020817547579635543648966232, −3.95451174510743665402743915485, −3.18247012507387119566809696750, −2.32964488262253205424706407156, −0.51916110966118557047575058199,
0.51916110966118557047575058199, 2.32964488262253205424706407156, 3.18247012507387119566809696750, 3.95451174510743665402743915485, 4.86020817547579635543648966232, 6.02684210173332234104860151617, 6.27574617805035798262057908141, 7.57960106576853046343584865094, 8.091725484064501883319978493826, 8.564530664989272348404745281383