| L(s) = 1 | + 3-s − 4·5-s − 7-s + 9-s + 4·11-s + 2·13-s − 4·15-s − 4·17-s − 19-s − 21-s + 11·25-s + 27-s + 4·29-s − 8·31-s + 4·33-s + 4·35-s − 2·37-s + 2·39-s − 6·41-s + 12·43-s − 4·45-s − 10·47-s + 49-s − 4·51-s − 16·55-s − 57-s − 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 1.03·15-s − 0.970·17-s − 0.229·19-s − 0.218·21-s + 11/5·25-s + 0.192·27-s + 0.742·29-s − 1.43·31-s + 0.696·33-s + 0.676·35-s − 0.328·37-s + 0.320·39-s − 0.937·41-s + 1.82·43-s − 0.596·45-s − 1.45·47-s + 1/7·49-s − 0.560·51-s − 2.15·55-s − 0.132·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.302068934264645221653868220062, −7.64754005835707289818431917192, −6.84999749024954386478888405215, −6.37316920291720248191233554695, −4.94857969994425135364560553645, −4.01272932648552777073478051830, −3.76967054401044198868817204495, −2.81855505689987175936276548328, −1.40345885206506099991532976356, 0,
1.40345885206506099991532976356, 2.81855505689987175936276548328, 3.76967054401044198868817204495, 4.01272932648552777073478051830, 4.94857969994425135364560553645, 6.37316920291720248191233554695, 6.84999749024954386478888405215, 7.64754005835707289818431917192, 8.302068934264645221653868220062
