L(s) = 1 | − 5-s − 7-s − 4·13-s − 6·17-s − 2·19-s + 6·23-s + 25-s − 2·31-s + 35-s + 2·37-s − 6·41-s + 4·43-s + 49-s + 6·53-s + 12·59-s − 10·61-s + 4·65-s + 4·67-s + 12·71-s − 4·73-s − 8·79-s + 12·83-s + 6·85-s − 6·89-s + 4·91-s + 2·95-s + 8·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.10·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 0.359·31-s + 0.169·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s + 1/7·49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s + 0.496·65-s + 0.488·67-s + 1.42·71-s − 0.468·73-s − 0.900·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.419·91-s + 0.205·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.105415283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.105415283\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.312417494085986509106620907657, −7.32299067164021900893988658054, −6.95690613180478001261141513162, −6.19263020239855182713744341321, −5.16406092827354928586251405384, −4.58279289915766210559272218207, −3.76718008701211949698418282941, −2.80701696291884140593044580997, −2.05631859112990099961601757225, −0.54790171512367851136938199577,
0.54790171512367851136938199577, 2.05631859112990099961601757225, 2.80701696291884140593044580997, 3.76718008701211949698418282941, 4.58279289915766210559272218207, 5.16406092827354928586251405384, 6.19263020239855182713744341321, 6.95690613180478001261141513162, 7.32299067164021900893988658054, 8.312417494085986509106620907657