| L(s) = 1 | − 5-s + 6·11-s + 6·17-s − 4·19-s − 6·23-s + 25-s + 2·29-s + 8·31-s − 2·37-s + 10·41-s − 12·43-s + 8·47-s + 2·53-s − 6·55-s + 4·59-s + 8·61-s − 16·67-s + 10·71-s + 4·79-s − 4·83-s − 6·85-s − 6·89-s + 4·95-s + 8·97-s − 10·101-s + 12·103-s + 2·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.80·11-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s + 1.56·41-s − 1.82·43-s + 1.16·47-s + 0.274·53-s − 0.809·55-s + 0.520·59-s + 1.02·61-s − 1.95·67-s + 1.18·71-s + 0.450·79-s − 0.439·83-s − 0.650·85-s − 0.635·89-s + 0.410·95-s + 0.812·97-s − 0.995·101-s + 1.18·103-s + 0.193·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.208972365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.208972365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.79616168989266321096892758940, −7.03901316757218195021878650974, −6.34469530383360817550283961531, −5.89522331052163953386410979547, −4.86178556809419532220678680653, −4.05413221068059406057583353522, −3.70534356545432874739043622086, −2.68767366528519314978728635855, −1.61331065700080997911850805410, −0.76653707219121105320473380665,
0.76653707219121105320473380665, 1.61331065700080997911850805410, 2.68767366528519314978728635855, 3.70534356545432874739043622086, 4.05413221068059406057583353522, 4.86178556809419532220678680653, 5.89522331052163953386410979547, 6.34469530383360817550283961531, 7.03901316757218195021878650974, 7.79616168989266321096892758940
