| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 2·9-s − 10-s + 6·11-s − 12-s − 5·13-s + 15-s + 16-s − 2·18-s − 3·19-s − 20-s + 6·22-s + 2·23-s − 24-s + 25-s − 5·26-s + 5·27-s + 29-s + 30-s + 7·31-s + 32-s − 6·33-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 − 2/3·9-s − 0.316·10-s + 1.80·11-s − 0.288·12-s − 1.38·13-s + 0.258·15-s + 1/4·16-s − 0.471·18-s − 0.688·19-s − 0.223·20-s + 1.27·22-s + 0.417·23-s − 0.204·24-s + 1/5·25-s − 0.980·26-s + 0.962·27-s + 0.185·29-s + 0.182·30-s + 1.25·31-s + 0.176·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.036554560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036554560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.712002507463661045949468399659, −7.950530341915683838966445632915, −6.85788309641515308707058303973, −6.56734828986171089391735239058, −5.67452402333251807305592667484, −4.77397936232840205266439013192, −4.22490475105922805237759369683, −3.22956647387255580327683897196, −2.26080068651121145132556526407, −0.811327097822202599534580088942,
0.811327097822202599534580088942, 2.26080068651121145132556526407, 3.22956647387255580327683897196, 4.22490475105922805237759369683, 4.77397936232840205266439013192, 5.67452402333251807305592667484, 6.56734828986171089391735239058, 6.85788309641515308707058303973, 7.950530341915683838966445632915, 8.712002507463661045949468399659
