L(s) = 1 | + 5-s − 7-s + 2·13-s + 6·17-s + 4·19-s + 25-s + 6·29-s + 4·31-s − 35-s + 2·37-s − 6·41-s − 8·43-s − 12·47-s + 49-s − 6·53-s − 12·59-s + 2·61-s + 2·65-s − 8·67-s + 14·73-s + 16·79-s + 12·83-s + 6·85-s − 6·89-s − 2·91-s + 4·95-s + 14·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + 1.63·73-s + 1.80·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.209·91-s + 0.410·95-s + 1.42·97-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
\(L(1)\) |
\(\approx\) |
\(2.298344657\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.298344657\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.128512562785316186820185573307, −7.66768959607577836840728490346, −6.55557068677345152926648458162, −6.24912364541800254645460865735, −5.26492371562926131202981843971, −4.73638158563715827960959455579, −3.39643631794807687387241543504, −3.13228179981233595332041390622, −1.79627847670602904159006500148, −0.867875508818869649803326174881,
0.867875508818869649803326174881, 1.79627847670602904159006500148, 3.13228179981233595332041390622, 3.39643631794807687387241543504, 4.73638158563715827960959455579, 5.26492371562926131202981843971, 6.24912364541800254645460865735, 6.55557068677345152926648458162, 7.66768959607577836840728490346, 8.128512562785316186820185573307