L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 11-s + 12-s + 7·13-s + 15-s + 16-s − 4·17-s + 18-s + 19-s + 20-s − 22-s + 23-s + 24-s + 25-s + 7·26-s + 27-s − 8·29-s + 30-s + 6·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s + 1.94·13-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.213·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 1.37·26-s + 0.192·27-s − 1.48·29-s + 0.182·30-s + 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.679844049\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.679844049\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.397658937382439861120277010335, −8.696067431211012704534067965923, −7.948349276844601853110894154905, −6.90671068971383230419727902926, −6.18513610864573746778108798813, −5.39546449121280347597416973893, −4.28153743220883784627096317325, −3.52400219693419364368205032145, −2.50614568352784799762211898634, −1.41043715253788005963268142129,
1.41043715253788005963268142129, 2.50614568352784799762211898634, 3.52400219693419364368205032145, 4.28153743220883784627096317325, 5.39546449121280347597416973893, 6.18513610864573746778108798813, 6.90671068971383230419727902926, 7.948349276844601853110894154905, 8.696067431211012704534067965923, 9.397658937382439861120277010335