| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 11-s + 12-s + 7·13-s + 16-s − 4·17-s − 18-s − 19-s + 22-s − 23-s − 24-s − 7·26-s + 27-s − 8·29-s − 6·31-s − 32-s − 33-s + 4·34-s + 36-s + 3·37-s + 38-s + 7·39-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 − 0.301·11-s + 0.288·12-s + 1.94·13-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.229·19-s + 0.213·22-s − 0.208·23-s − 0.204·24-s − 1.37·26-s + 0.192·27-s − 1.48·29-s − 1.07·31-s − 0.176·32-s − 0.174·33-s + 0.685·34-s + 1/6·36-s + 0.493·37-s + 0.162·38-s + 1.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 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
−7.67555152242832042954535491516, −7.05525874007674222868396515897, −6.20362615024844725641955867669, −5.73966122871758072512789883779, −4.56735096282531774427030346268, −3.75210721650779885118568615299, −3.12886905694715598722221979193, −2.03299936806186887897303950642, −1.41754065486666937185517705561, 0,
1.41754065486666937185517705561, 2.03299936806186887897303950642, 3.12886905694715598722221979193, 3.75210721650779885118568615299, 4.56735096282531774427030346268, 5.73966122871758072512789883779, 6.20362615024844725641955867669, 7.05525874007674222868396515897, 7.67555152242832042954535491516
