| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s − 3·7-s + 3·8-s + 9-s + 10-s + 5·11-s + 12-s − 4·13-s + 3·14-s + 15-s − 16-s − 2·17-s − 18-s + 19-s + 20-s + 3·21-s − 5·22-s + 8·23-s − 3·24-s − 4·25-s + 4·26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 1.50·11-s + 0.288·12-s − 1.10·13-s + 0.801·14-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.654·21-s − 1.06·22-s + 1.66·23-s − 0.612·24-s − 4/5·25-s + 0.784·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160113 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160113 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8089879216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8089879216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.13610127168032, −12.82158625870667, −12.30137959385350, −11.83521548659977, −11.31710249021504, −10.89396154190144, −10.36023448062668, −9.659298532183777, −9.517701067723166, −9.104181465017195, −8.686265005780842, −7.902284078136918, −7.350433441356979, −7.029295769107462, −6.595695574685567, −5.964589907639733, −5.322609922107331, −4.806665533944189, −4.249000569628443, −3.712961140765483, −3.321079845819145, −2.361839883236419, −1.685473601659260, −0.8023299131059619, −0.4469964996050015,
0.4469964996050015, 0.8023299131059619, 1.685473601659260, 2.361839883236419, 3.321079845819145, 3.712961140765483, 4.249000569628443, 4.806665533944189, 5.322609922107331, 5.964589907639733, 6.595695574685567, 7.029295769107462, 7.350433441356979, 7.902284078136918, 8.686265005780842, 9.104181465017195, 9.517701067723166, 9.659298532183777, 10.36023448062668, 10.89396154190144, 11.31710249021504, 11.83521548659977, 12.30137959385350, 12.82158625870667, 13.13610127168032
