| L(s) = 1 | + 3-s + 5-s − 2·9-s − 5·11-s + 5·13-s + 15-s + 5·17-s + 8·23-s + 25-s − 5·27-s + 29-s − 2·31-s − 5·33-s − 4·37-s + 5·39-s − 2·41-s − 4·43-s − 2·45-s − 13·47-s + 5·51-s + 8·53-s − 5·55-s + 4·59-s − 2·61-s + 5·65-s + 8·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s + 1.38·13-s + 0.258·15-s + 1.21·17-s + 1.66·23-s + 1/5·25-s − 0.962·27-s + 0.185·29-s − 0.359·31-s − 0.870·33-s − 0.657·37-s + 0.800·39-s − 0.312·41-s − 0.609·43-s − 0.298·45-s − 1.89·47-s + 0.700·51-s + 1.09·53-s − 0.674·55-s + 0.520·59-s − 0.256·61-s + 0.620·65-s + 0.977·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.742266210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.742266210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−16.11888158009216, −15.25949679665744, −14.92857892968620, −14.33858150529363, −13.65936215215326, −13.27047840955024, −12.95107742162466, −12.10277630000957, −11.39854439388291, −10.86312718594222, −10.35432724186378, −9.764064358785734, −8.983631518284694, −8.568799278199317, −8.015303184853982, −7.468372628239670, −6.613344249478544, −5.958036313646995, −5.230475579599218, −4.989202418176694, −3.549080842226415, −3.303634815607494, −2.549854795366001, −1.698651711671920, −0.6951731787083527,
0.6951731787083527, 1.698651711671920, 2.549854795366001, 3.303634815607494, 3.549080842226415, 4.989202418176694, 5.230475579599218, 5.958036313646995, 6.613344249478544, 7.468372628239670, 8.015303184853982, 8.568799278199317, 8.983631518284694, 9.764064358785734, 10.35432724186378, 10.86312718594222, 11.39854439388291, 12.10277630000957, 12.95107742162466, 13.27047840955024, 13.65936215215326, 14.33858150529363, 14.92857892968620, 15.25949679665744, 16.11888158009216
