| L(s) = 1 | + 5-s + 3·7-s + 3·13-s + 17-s + 5·19-s + 6·23-s + 25-s + 29-s + 3·35-s + 9·37-s + 4·41-s + 4·43-s − 4·47-s + 2·49-s + 2·53-s − 8·59-s − 4·61-s + 3·65-s − 8·67-s + 5·71-s + 4·73-s + 4·79-s + 15·83-s + 85-s + 6·89-s + 9·91-s + 5·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.832·13-s + 0.242·17-s + 1.14·19-s + 1.25·23-s + 1/5·25-s + 0.185·29-s + 0.507·35-s + 1.47·37-s + 0.624·41-s + 0.609·43-s − 0.583·47-s + 2/7·49-s + 0.274·53-s − 1.04·59-s − 0.512·61-s + 0.372·65-s − 0.977·67-s + 0.593·71-s + 0.468·73-s + 0.450·79-s + 1.64·83-s + 0.108·85-s + 0.635·89-s + 0.943·91-s + 0.512·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.724645665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.724645665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.85705318696376, −13.42883778719926, −13.12344086930638, −12.22164945264664, −12.07061276908293, −11.20394293549698, −10.99414770780407, −10.67531371695328, −9.762883649059776, −9.413438003827844, −8.965573093357766, −8.278641522854248, −7.851701378140654, −7.439287229357945, −6.714115592325925, −6.176729422780069, −5.573824985216320, −5.127875397233871, −4.583579183452137, −4.000580299415600, −3.173573097189325, −2.739348033803337, −1.871832414739263, −1.240198886775938, −0.7928656114944453,
0.7928656114944453, 1.240198886775938, 1.871832414739263, 2.739348033803337, 3.173573097189325, 4.000580299415600, 4.583579183452137, 5.127875397233871, 5.573824985216320, 6.176729422780069, 6.714115592325925, 7.439287229357945, 7.851701378140654, 8.278641522854248, 8.965573093357766, 9.413438003827844, 9.762883649059776, 10.67531371695328, 10.99414770780407, 11.20394293549698, 12.07061276908293, 12.22164945264664, 13.12344086930638, 13.42883778719926, 13.85705318696376
