L(s) = 1 | + 5-s + 3·11-s + 4·13-s + 4·19-s + 8·23-s − 4·25-s + 3·29-s + 5·31-s + 8·37-s − 8·41-s − 6·43-s + 10·47-s − 9·53-s + 3·55-s − 5·59-s − 10·61-s + 4·65-s − 6·67-s + 10·71-s + 2·73-s − 11·79-s + 7·83-s + 18·89-s + 4·95-s − 17·97-s + 2·101-s − 11·107-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.904·11-s + 1.10·13-s + 0.917·19-s + 1.66·23-s − 4/5·25-s + 0.557·29-s + 0.898·31-s + 1.31·37-s − 1.24·41-s − 0.914·43-s + 1.45·47-s − 1.23·53-s + 0.404·55-s − 0.650·59-s − 1.28·61-s + 0.496·65-s − 0.733·67-s + 1.18·71-s + 0.234·73-s − 1.23·79-s + 0.768·83-s + 1.90·89-s + 0.410·95-s − 1.72·97-s + 0.199·101-s − 1.06·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.837167303\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.837167303\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.998157978762509709012849864254, −7.16183400955335543876118219905, −6.44162281874474764302538262339, −5.99193522513282755009195665126, −5.11162449121302522698816434748, −4.38619854192959487427000112591, −3.47925632125466026069668696822, −2.84811242361592356592684674685, −1.61004889127807801694244292086, −0.948672759887688732959170989133,
0.948672759887688732959170989133, 1.61004889127807801694244292086, 2.84811242361592356592684674685, 3.47925632125466026069668696822, 4.38619854192959487427000112591, 5.11162449121302522698816434748, 5.99193522513282755009195665126, 6.44162281874474764302538262339, 7.16183400955335543876118219905, 7.998157978762509709012849864254