| L(s) = 1 | + 7-s − 3·9-s − 4·13-s + 4·17-s − 4·19-s + 8·23-s − 2·29-s − 8·31-s + 8·37-s + 6·41-s − 8·43-s + 8·47-s + 49-s + 4·59-s + 6·61-s − 3·63-s − 8·67-s + 12·71-s − 4·73-s − 4·79-s + 9·81-s − 10·89-s − 4·91-s − 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s − 1.10·13-s + 0.970·17-s − 0.917·19-s + 1.66·23-s − 0.371·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.520·59-s + 0.768·61-s − 0.377·63-s − 0.977·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s + 81-s − 1.05·89-s − 0.419·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−16.68084948210589, −16.59090436556192, −15.42702571615796, −14.89129794700684, −14.53573618647659, −14.19331535379940, −13.11352682380036, −12.90038884443079, −12.07945848818981, −11.61115286190337, −10.91170705003937, −10.57728709189699, −9.575922578630290, −9.230365739196304, −8.482721349125695, −7.885535783601306, −7.268022461145143, −6.645665291628469, −5.633681797976779, −5.369728025296785, −4.555149650781371, −3.727863904195598, −2.838761430326467, −2.299284735085640, −1.146745363322601, 0,
1.146745363322601, 2.299284735085640, 2.838761430326467, 3.727863904195598, 4.555149650781371, 5.369728025296785, 5.633681797976779, 6.645665291628469, 7.268022461145143, 7.885535783601306, 8.482721349125695, 9.230365739196304, 9.575922578630290, 10.57728709189699, 10.91170705003937, 11.61115286190337, 12.07945848818981, 12.90038884443079, 13.11352682380036, 14.19331535379940, 14.53573618647659, 14.89129794700684, 15.42702571615796, 16.59090436556192, 16.68084948210589
