| L(s) = 1 | + 20·7-s + 70·13-s − 56·19-s − 125·25-s + 308·31-s − 110·37-s + 520·43-s + 57·49-s − 182·61-s + 880·67-s + 1.19e3·73-s + 884·79-s + 1.40e3·91-s − 1.33e3·97-s + 1.82e3·103-s + 646·109-s + ⋯ |
| L(s) = 1 | + 1.07·7-s + 1.49·13-s − 0.676·19-s − 25-s + 1.78·31-s − 0.488·37-s + 1.84·43-s + 0.166·49-s − 0.382·61-s + 1.60·67-s + 1.90·73-s + 1.25·79-s + 1.61·91-s − 1.39·97-s + 1.74·103-s + 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.418196208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.418196208\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 308 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 - 520 T + p^{3} T^{2} \) |
| 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 | \( 1 - 880 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 1190 T + p^{3} T^{2} \) |
| 79 | \( 1 - 884 T + p^{3} T^{2} \) |
| 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
−10.51687634246492253245905967076, −9.371759346978080464148608284740, −8.373647243049877696384062075772, −7.927055371481811556556113521003, −6.61167720394527182598591035150, −5.76395383888221181170247647998, −4.63194166505924051646639150145, −3.71530891672760758103454863265, −2.19280022880392417896749201987, −0.997372270815512212763211946172,
0.997372270815512212763211946172, 2.19280022880392417896749201987, 3.71530891672760758103454863265, 4.63194166505924051646639150145, 5.76395383888221181170247647998, 6.61167720394527182598591035150, 7.927055371481811556556113521003, 8.373647243049877696384062075772, 9.371759346978080464148608284740, 10.51687634246492253245905967076
