L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2·7-s − 8-s + 9-s − 10-s − 4·11-s + 12-s + 3·13-s + 2·14-s + 15-s + 16-s − 18-s − 19-s + 20-s − 2·21-s + 4·22-s − 23-s − 24-s + 25-s − 3·26-s + 27-s − 2·28-s − 4·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.832·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.229·19-s + 0.223·20-s − 0.436·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.192·27-s − 0.377·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−13.37255316690772, −13.10850189729995, −12.66922052718583, −12.08807808283801, −11.46749957912737, −11.00736366205709, −10.37189113050438, −10.14417061154727, −9.714518263559273, −9.186335856352686, −8.664703825417329, −8.191986480064250, −7.950850295335829, −7.135411758474739, −6.807939633198193, −6.194437418532912, −5.803724047112656, −5.099121841828019, −4.593811815491292, −3.699636630516757, −3.193912964023998, −2.889899195692310, −1.959761105844872, −1.764950801366867, −0.7407563697215129, 0,
0.7407563697215129, 1.764950801366867, 1.959761105844872, 2.889899195692310, 3.193912964023998, 3.699636630516757, 4.593811815491292, 5.099121841828019, 5.803724047112656, 6.194437418532912, 6.807939633198193, 7.135411758474739, 7.950850295335829, 8.191986480064250, 8.664703825417329, 9.186335856352686, 9.714518263559273, 10.14417061154727, 10.37189113050438, 11.00736366205709, 11.46749957912737, 12.08807808283801, 12.66922052718583, 13.10850189729995, 13.37255316690772