L(s) = 1 | + 2·7-s + 4·11-s + 6·13-s − 6·17-s − 4·23-s − 5·25-s − 4·29-s + 10·31-s + 2·37-s + 2·41-s + 8·43-s + 12·47-s − 3·49-s + 12·53-s + 4·59-s + 2·61-s + 4·67-s + 4·71-s − 10·73-s + 8·77-s − 6·79-s − 12·83-s − 2·89-s + 12·91-s − 6·97-s − 4·101-s − 10·103-s + ⋯ |
L(s) = 1 | + 0.755·7-s + 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.834·23-s − 25-s − 0.742·29-s + 1.79·31-s + 0.328·37-s + 0.312·41-s + 1.21·43-s + 1.75·47-s − 3/7·49-s + 1.64·53-s + 0.520·59-s + 0.256·61-s + 0.488·67-s + 0.474·71-s − 1.17·73-s + 0.911·77-s − 0.675·79-s − 1.31·83-s − 0.211·89-s + 1.25·91-s − 0.609·97-s − 0.398·101-s − 0.985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.958510323\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.958510323\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−9.732093454641320086632953897344, −8.788616180700843175291693061294, −8.396078230698006089481678801982, −7.31199108027851415208504538208, −6.32971218305453945925176285036, −5.76550563678950678115455651188, −4.27440644248673667608699010194, −3.95147465503398880690094499828, −2.30530051971351607466902904067, −1.16901968292393942798169217496,
1.16901968292393942798169217496, 2.30530051971351607466902904067, 3.95147465503398880690094499828, 4.27440644248673667608699010194, 5.76550563678950678115455651188, 6.32971218305453945925176285036, 7.31199108027851415208504538208, 8.396078230698006089481678801982, 8.788616180700843175291693061294, 9.732093454641320086632953897344