| L(s) = 1 | + (−1 − i)2-s + 2i·4-s − 2·5-s + (2 − 2i)8-s + (2 + 2i)10-s − 4·11-s + (−3 − 2i)13-s − 4·16-s + 6·17-s + 6·19-s − 4i·20-s + (4 + 4i)22-s − 25-s + (1 + 5i)26-s + 6i·29-s + ⋯ |
| L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s − 0.894·5-s + (0.707 − 0.707i)8-s + (0.632 + 0.632i)10-s − 1.20·11-s + (−0.832 − 0.554i)13-s − 16-s + 1.45·17-s + 1.37·19-s − 0.894i·20-s + (0.852 + 0.852i)22-s − 0.200·25-s + (0.196 + 0.980i)26-s + 1.11i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.702967 + 0.0696074i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.702967 + 0.0696074i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2iT - 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 - 4iT - 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−9.968593741110944114633720100437, −9.556463765132434087575980456084, −8.211732835842189818607698946292, −7.74872946903207527096974164902, −7.25951388435868742764990445663, −5.59694657828558710794521876419, −4.63805828961142036520983252238, −3.36629033523467499502947819692, −2.74233793767591247331190600866, −0.974158318842557363151495598103,
0.55438768228684144843629829877, 2.35223970780657332798692932164, 3.78945020359951230579920482569, 5.07234849171507691328590934684, 5.63369870355802609696417928750, 6.97468700163958322322582461406, 7.73115268382516386041871556911, 7.971110343137239931229308450546, 9.190455465490961778464599234160, 9.956396021289757611160850726061
