L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s + 14-s + 15-s + 16-s − 5·17-s + 18-s + 2·19-s + 20-s + 21-s + 22-s − 9·23-s + 24-s − 4·25-s − 26-s + 27-s + 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 + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.218·21-s + 0.213·22-s − 1.87·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 41 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−15.67100865451465, −15.21461737025624, −14.50161996903482, −14.08038079341017, −13.83480106327505, −13.11397029092343, −12.73394115853492, −11.88314673598639, −11.68364707333980, −10.83297571818131, −10.37565683584470, −9.630538644563460, −9.220239792792816, −8.534066894447573, −7.774219763508119, −7.469604836446753, −6.562700308254817, −6.135546157415835, −5.430784205027964, −4.755126872227921, −4.072948453215712, −3.622901340704249, −2.665068214718699, −2.014646561038108, −1.551952636484398, 0,
1.551952636484398, 2.014646561038108, 2.665068214718699, 3.622901340704249, 4.072948453215712, 4.755126872227921, 5.430784205027964, 6.135546157415835, 6.562700308254817, 7.469604836446753, 7.774219763508119, 8.534066894447573, 9.220239792792816, 9.630538644563460, 10.37565683584470, 10.83297571818131, 11.68364707333980, 11.88314673598639, 12.73394115853492, 13.11397029092343, 13.83480106327505, 14.08038079341017, 14.50161996903482, 15.21461737025624, 15.67100865451465