| L(s) = 1 | − 3-s − 2·5-s − 2·9-s + 4·11-s − 2·13-s + 2·15-s − 6·17-s − 19-s − 6·23-s − 25-s + 5·27-s − 4·29-s − 8·31-s − 4·33-s − 2·37-s + 2·39-s + 2·43-s + 4·45-s − 7·47-s − 7·49-s + 6·51-s − 9·53-s − 8·55-s + 57-s − 5·59-s + 2·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 2/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.229·19-s − 1.25·23-s − 1/5·25-s + 0.962·27-s − 0.742·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.320·39-s + 0.304·43-s + 0.596·45-s − 1.02·47-s − 49-s + 0.840·51-s − 1.23·53-s − 1.07·55-s + 0.132·57-s − 0.650·59-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5077 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.21488571406156, −15.78086722210985, −14.98154795186065, −14.66193598174094, −14.14264064865971, −13.45231920024546, −12.75654961845194, −12.20492694940360, −11.74932643715353, −11.23165793477833, −11.01351252635053, −10.15394732913426, −9.374949128418145, −8.982161881371317, −8.314396077357754, −7.700237684938396, −7.088716035777523, −6.356793478701476, −6.054150153963881, −5.124624424357814, −4.516274479956345, −3.892619242829995, −3.338418945829633, −2.268602176101206, −1.551607409521583, 0, 0,
1.551607409521583, 2.268602176101206, 3.338418945829633, 3.892619242829995, 4.516274479956345, 5.124624424357814, 6.054150153963881, 6.356793478701476, 7.088716035777523, 7.700237684938396, 8.314396077357754, 8.982161881371317, 9.374949128418145, 10.15394732913426, 11.01351252635053, 11.23165793477833, 11.74932643715353, 12.20492694940360, 12.75654961845194, 13.45231920024546, 14.14264064865971, 14.66193598174094, 14.98154795186065, 15.78086722210985, 16.21488571406156
