| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 4·11-s + 12-s + 2·13-s + 16-s − 17-s − 18-s + 4·19-s − 4·22-s − 24-s − 2·26-s + 27-s − 2·29-s + 8·31-s − 32-s + 4·33-s + 34-s + 36-s − 6·37-s − 4·38-s + 2·39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.917·19-s − 0.852·22-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.696·33-s + 0.171·34-s + 1/6·36-s − 0.986·37-s − 0.648·38-s + 0.320·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.861291521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.861291521\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.891511613917682416170599280455, −8.310462568854645808738516053588, −7.48010619200380663499134996167, −6.73690075819311214031311678822, −6.08439557059290044397198605751, −4.92646119617734323831892460778, −3.85183185671284510497638778861, −3.13544879646318674799993075010, −1.94324442746163824191174281871, −0.993709209615856864448811822787,
0.993709209615856864448811822787, 1.94324442746163824191174281871, 3.13544879646318674799993075010, 3.85183185671284510497638778861, 4.92646119617734323831892460778, 6.08439557059290044397198605751, 6.73690075819311214031311678822, 7.48010619200380663499134996167, 8.310462568854645808738516053588, 8.891511613917682416170599280455
