| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 5·11-s + 5·13-s − 14-s + 16-s + 3·17-s − 6·19-s + 5·22-s + 23-s − 5·26-s + 28-s − 29-s + 31-s − 32-s − 3·34-s + 10·37-s + 6·38-s − 4·41-s − 5·43-s − 5·44-s − 46-s − 11·47-s + 49-s + 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 1.50·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.37·19-s + 1.06·22-s + 0.208·23-s − 0.980·26-s + 0.188·28-s − 0.185·29-s + 0.179·31-s − 0.176·32-s − 0.514·34-s + 1.64·37-s + 0.973·38-s − 0.624·41-s − 0.762·43-s − 0.753·44-s − 0.147·46-s − 1.60·47-s + 1/7·49-s + 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.73313138465549144314314482715, −6.61717005645662952678147858187, −6.16968511963822287793593036436, −5.36274024516048329195387658077, −4.66892978878418267090038752492, −3.68736314986623766863902524606, −2.90473882737923846156944983181, −2.05902736145126513421362801275, −1.17551343553520906890445343767, 0,
1.17551343553520906890445343767, 2.05902736145126513421362801275, 2.90473882737923846156944983181, 3.68736314986623766863902524606, 4.66892978878418267090038752492, 5.36274024516048329195387658077, 6.16968511963822287793593036436, 6.61717005645662952678147858187, 7.73313138465549144314314482715
