| L(s) = 1 | + 3-s − 4·5-s + 9-s − 2·11-s − 4·15-s + 2·17-s + 8·19-s − 4·23-s + 11·25-s + 27-s − 6·29-s − 4·31-s − 2·33-s − 6·37-s + 12·41-s − 4·43-s − 4·45-s − 6·47-s − 7·49-s + 2·51-s − 2·53-s + 8·55-s + 8·57-s − 14·59-s + 10·61-s − 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1/3·9-s − 0.603·11-s − 1.03·15-s + 0.485·17-s + 1.83·19-s − 0.834·23-s + 11/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s + 1.87·41-s − 0.609·43-s − 0.596·45-s − 0.875·47-s − 49-s + 0.280·51-s − 0.274·53-s + 1.07·55-s + 1.05·57-s − 1.82·59-s + 1.28·61-s − 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316128985\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316128985\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.86316522899630954788479439305, −7.46086347248797051903138725260, −6.71852525997562721644290256948, −5.57797429307990342218329243413, −4.94284655596875632093269264489, −4.09186280433964370332575554981, −3.42908361296580903109125883874, −3.03676069906676700119323436080, −1.76116852909308634429700522403, −0.54643906041975952555003326483,
0.54643906041975952555003326483, 1.76116852909308634429700522403, 3.03676069906676700119323436080, 3.42908361296580903109125883874, 4.09186280433964370332575554981, 4.94284655596875632093269264489, 5.57797429307990342218329243413, 6.71852525997562721644290256948, 7.46086347248797051903138725260, 7.86316522899630954788479439305
