| L(s) = 1 | − 3-s + 9-s − 4·11-s − 6·13-s + 6·17-s + 4·19-s + 4·23-s − 27-s − 2·29-s − 8·31-s + 4·33-s − 6·37-s + 6·39-s − 6·41-s − 8·43-s − 6·51-s − 6·53-s − 4·57-s + 4·59-s − 10·61-s − 8·67-s − 4·69-s + 12·71-s − 14·73-s − 16·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.960·39-s − 0.937·41-s − 1.21·43-s − 0.840·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.977·67-s − 0.481·69-s + 1.42·71-s − 1.63·73-s − 1.80·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06423054812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06423054812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.52577873976342, −12.98488088342883, −12.52576869644812, −12.24339145320795, −11.61825787496618, −11.30197347931408, −10.48133695121823, −10.27469904548658, −9.777785895434254, −9.358685813627426, −8.667220452875893, −7.979722223362035, −7.525414047653866, −7.198298721432275, −6.766973246232646, −5.750946349047155, −5.453319906812807, −5.084204574617922, −4.655774371015079, −3.743846135350126, −3.060485862559943, −2.786394407577179, −1.779194311936586, −1.299044055923995, −0.08017919023929260,
0.08017919023929260, 1.299044055923995, 1.779194311936586, 2.786394407577179, 3.060485862559943, 3.743846135350126, 4.655774371015079, 5.084204574617922, 5.453319906812807, 5.750946349047155, 6.766973246232646, 7.198298721432275, 7.525414047653866, 7.979722223362035, 8.667220452875893, 9.358685813627426, 9.777785895434254, 10.27469904548658, 10.48133695121823, 11.30197347931408, 11.61825787496618, 12.24339145320795, 12.52576869644812, 12.98488088342883, 13.52577873976342
