L(s) = 1 | − 3-s + 5-s − 7-s + 9-s + 4·11-s + 6·13-s − 15-s − 2·17-s + 8·19-s + 21-s + 4·23-s + 25-s − 27-s − 6·29-s + 4·31-s − 4·33-s − 35-s + 2·37-s − 6·39-s − 2·41-s + 12·43-s + 45-s + 49-s + 2·51-s − 2·53-s + 4·55-s − 8·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s − 0.485·17-s + 1.83·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 0.169·35-s + 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.82·43-s + 0.149·45-s + 1/7·49-s + 0.280·51-s − 0.274·53-s + 0.539·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.330003541\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.330003541\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−7.945694647467123497632967591632, −7.02441805968521416558850437535, −6.58013144075984850837983148316, −5.83348111807032833453365810096, −5.40193968655204848246285151068, −4.29207768501800499210453217765, −3.68344655376293356705460126216, −2.83602530293778452714344445391, −1.48792866077805377526165091358, −0.922323774977906685673954559229,
0.922323774977906685673954559229, 1.48792866077805377526165091358, 2.83602530293778452714344445391, 3.68344655376293356705460126216, 4.29207768501800499210453217765, 5.40193968655204848246285151068, 5.83348111807032833453365810096, 6.58013144075984850837983148316, 7.02441805968521416558850437535, 7.945694647467123497632967591632