| L(s) = 1 | − 2·3-s − 5-s + 9-s − 5·11-s + 2·13-s + 2·15-s − 2·17-s + 4·23-s + 25-s + 4·27-s − 9·29-s − 5·31-s + 10·33-s + 4·37-s − 4·39-s − 10·41-s + 8·43-s − 45-s + 2·47-s − 7·49-s + 4·51-s − 6·53-s + 5·55-s + 7·59-s − 7·61-s − 2·65-s − 14·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.516·15-s − 0.485·17-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 1.67·29-s − 0.898·31-s + 1.74·33-s + 0.657·37-s − 0.640·39-s − 1.56·41-s + 1.21·43-s − 0.149·45-s + 0.291·47-s − 49-s + 0.560·51-s − 0.824·53-s + 0.674·55-s + 0.911·59-s − 0.896·61-s − 0.248·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2402297254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2402297254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−15.37426505376677, −14.78885916285860, −14.13975511668146, −13.33885093018338, −12.92843809567932, −12.66542069968950, −11.82084785526342, −11.32428866004763, −10.97354453175376, −10.59921457409179, −9.962213394220181, −9.137746652158870, −8.682496168036965, −7.902948130821965, −7.454169561742373, −6.864072917678048, −6.091239902653983, −5.673464028189557, −5.042944829259831, −4.638319718903017, −3.718676384925135, −3.071625927084915, −2.251903512087860, −1.285713445724446, −0.2128945391713597,
0.2128945391713597, 1.285713445724446, 2.251903512087860, 3.071625927084915, 3.718676384925135, 4.638319718903017, 5.042944829259831, 5.673464028189557, 6.091239902653983, 6.864072917678048, 7.454169561742373, 7.902948130821965, 8.682496168036965, 9.137746652158870, 9.962213394220181, 10.59921457409179, 10.97354453175376, 11.32428866004763, 11.82084785526342, 12.66542069968950, 12.92843809567932, 13.33885093018338, 14.13975511668146, 14.78885916285860, 15.37426505376677
