| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 4·7-s − 8-s + 9-s + 2·11-s + 12-s + 2·13-s − 4·14-s + 16-s − 17-s − 18-s + 8·19-s + 4·21-s − 2·22-s + 23-s − 24-s − 2·26-s + 27-s + 4·28-s − 4·29-s − 2·31-s − 32-s + 2·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s + 0.872·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.755·28-s − 0.742·29-s − 0.359·31-s − 0.176·32-s + 0.348·33-s + 0.171·34-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
| \(L(1)\) |
\(\approx\) |
\(2.222835990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.222835990\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.842822966963033229580554033301, −8.195170028707783058854595973302, −7.56505200435682150316274640241, −6.95124268001693452408634245992, −5.80005067029779889254412140731, −4.99037744771183051401852881490, −3.99285406726243512113629587437, −3.03925960415236286645723483111, −1.82733199687905311423549882962, −1.15777355619552332379132537743,
1.15777355619552332379132537743, 1.82733199687905311423549882962, 3.03925960415236286645723483111, 3.99285406726243512113629587437, 4.99037744771183051401852881490, 5.80005067029779889254412140731, 6.95124268001693452408634245992, 7.56505200435682150316274640241, 8.195170028707783058854595973302, 8.842822966963033229580554033301
