| L(s) = 1 | − 89·13-s + 163·19-s − 125·25-s + 19·31-s − 433·37-s + 449·43-s − 182·61-s + 1.00e3·67-s + 919·73-s + 503·79-s + 1.33e3·97-s + 19·103-s − 1.56e3·109-s + ⋯ |
| L(s) = 1 | − 1.89·13-s + 1.96·19-s − 25-s + 0.110·31-s − 1.92·37-s + 1.59·43-s − 0.382·61-s + 1.83·67-s + 1.47·73-s + 0.716·79-s + 1.39·97-s + 0.0181·103-s − 1.37·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.685210177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685210177\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 19 T + p^{3} T^{2} \) |
| 37 | \( 1 + 433 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 449 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + 182 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 919 T + p^{3} T^{2} \) |
| 79 | \( 1 - 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1330 T + p^{3} T^{2} \) |
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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
−9.154806075944976789188518225551, −7.921151392326484619135831558284, −7.45135374706443790676030858379, −6.69399651915852671929246368760, −5.45505128713597931102470064235, −5.05125702570261115560106527906, −3.89214644767701940232365607956, −2.90616702434033013815363232700, −1.95523086906599417145683368441, −0.59932430017073904992039950005,
0.59932430017073904992039950005, 1.95523086906599417145683368441, 2.90616702434033013815363232700, 3.89214644767701940232365607956, 5.05125702570261115560106527906, 5.45505128713597931102470064235, 6.69399651915852671929246368760, 7.45135374706443790676030858379, 7.921151392326484619135831558284, 9.154806075944976789188518225551
