| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s + 11-s + 12-s − 13-s − 2·14-s + 16-s + 8·17-s + 18-s − 5·19-s − 2·21-s + 22-s + 9·23-s + 24-s − 26-s + 27-s − 2·28-s + 5·29-s + 7·31-s + 32-s + 33-s + 8·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 1.14·19-s − 0.436·21-s + 0.213·22-s + 1.87·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.928·29-s + 1.25·31-s + 0.176·32-s + 0.174·33-s + 1.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.293423188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.293423188\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.455301189446345553947352343026, −8.527758493988660530248485104186, −7.72098451440000952549385585429, −6.84516394586022105138556495825, −6.21095833028625901542898586868, −5.17018905881404842922243766259, −4.29801124091237918157127304665, −3.25269810672853646388462305223, −2.74409065102412383082160673869, −1.21584473992489391164356772872,
1.21584473992489391164356772872, 2.74409065102412383082160673869, 3.25269810672853646388462305223, 4.29801124091237918157127304665, 5.17018905881404842922243766259, 6.21095833028625901542898586868, 6.84516394586022105138556495825, 7.72098451440000952549385585429, 8.527758493988660530248485104186, 9.455301189446345553947352343026
