| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 11-s + 12-s + 3·13-s − 14-s + 16-s + 3·17-s + 18-s − 21-s + 22-s − 5·23-s + 24-s + 3·26-s + 27-s − 28-s − 7·29-s + 9·31-s + 32-s + 33-s + 3·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.218·21-s + 0.213·22-s − 1.04·23-s + 0.204·24-s + 0.588·26-s + 0.192·27-s − 0.188·28-s − 1.29·29-s + 1.61·31-s + 0.176·32-s + 0.174·33-s + 0.514·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.678778848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.678778848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−16.21674457138474, −15.85996671519890, −15.12010733144953, −14.77556337989223, −13.99347611750809, −13.73347808717820, −13.00970869552634, −12.67129752666731, −11.74622909804620, −11.53641822334352, −10.63797982765197, −9.946447340370678, −9.588676089392898, −8.694955613334117, −8.083812542973378, −7.603714426625758, −6.659249835894300, −6.262350715667930, −5.542609238101435, −4.751327101280031, −3.834144076222923, −3.589672153556698, −2.668115380795423, −1.886218138088275, −0.8869590185678108,
0.8869590185678108, 1.886218138088275, 2.668115380795423, 3.589672153556698, 3.834144076222923, 4.751327101280031, 5.542609238101435, 6.262350715667930, 6.659249835894300, 7.603714426625758, 8.083812542973378, 8.694955613334117, 9.588676089392898, 9.946447340370678, 10.63797982765197, 11.53641822334352, 11.74622909804620, 12.67129752666731, 13.00970869552634, 13.73347808717820, 13.99347611750809, 14.77556337989223, 15.12010733144953, 15.85996671519890, 16.21674457138474
