| L(s) = 1 | + 3-s − 3·7-s + 9-s + 2·11-s + 3·13-s + 6·17-s − 7·19-s − 3·21-s − 6·23-s + 27-s + 2·29-s + 5·31-s + 2·33-s − 10·37-s + 3·39-s + 12·41-s + 3·43-s + 10·47-s + 2·49-s + 6·51-s − 7·57-s − 6·59-s + 13·61-s − 3·63-s + 7·67-s − 6·69-s + 4·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.832·13-s + 1.45·17-s − 1.60·19-s − 0.654·21-s − 1.25·23-s + 0.192·27-s + 0.371·29-s + 0.898·31-s + 0.348·33-s − 1.64·37-s + 0.480·39-s + 1.87·41-s + 0.457·43-s + 1.45·47-s + 2/7·49-s + 0.840·51-s − 0.927·57-s − 0.781·59-s + 1.66·61-s − 0.377·63-s + 0.855·67-s − 0.722·69-s + 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.219535986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219535986\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.328581123207389939411321847736, −7.67187309952541547158695784134, −6.69931154349716503412960866300, −6.24638725213485855502889490405, −5.54092870319558395973689781087, −4.18883148172609513427426782540, −3.79736196515207346926195840996, −2.95797255925934707143477656324, −2.01685126964649461046898125664, −0.802772097749749091844725442220,
0.802772097749749091844725442220, 2.01685126964649461046898125664, 2.95797255925934707143477656324, 3.79736196515207346926195840996, 4.18883148172609513427426782540, 5.54092870319558395973689781087, 6.24638725213485855502889490405, 6.69931154349716503412960866300, 7.67187309952541547158695784134, 8.328581123207389939411321847736
