| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 4·11-s − 12-s + 13-s + 15-s + 16-s − 2·17-s − 18-s + 19-s − 20-s + 4·22-s + 4·23-s + 24-s + 25-s − 26-s − 27-s + 2·29-s − 30-s + 4·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-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 + 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9497547301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9497547301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.43130972577289, −12.13319370957224, −11.40479432257449, −11.03703705611502, −10.75039863975015, −10.42646268016144, −9.836113542828497, −9.261169882559280, −8.961757939336993, −8.390222104412487, −7.795541930275145, −7.610168151610912, −7.112705050235612, −6.437935815621637, −6.184302763625850, −5.546564018460733, −4.931069417070113, −4.723250951819014, −3.932537136403002, −3.405307374577011, −2.672996622164328, −2.418383556502703, −1.532045412083100, −0.8976159907259611, −0.3689728774547853,
0.3689728774547853, 0.8976159907259611, 1.532045412083100, 2.418383556502703, 2.672996622164328, 3.405307374577011, 3.932537136403002, 4.723250951819014, 4.931069417070113, 5.546564018460733, 6.184302763625850, 6.437935815621637, 7.112705050235612, 7.610168151610912, 7.795541930275145, 8.390222104412487, 8.961757939336993, 9.261169882559280, 9.836113542828497, 10.42646268016144, 10.75039863975015, 11.03703705611502, 11.40479432257449, 12.13319370957224, 12.43130972577289
