L(s) = 1 | + 5-s − 2·7-s − 2·13-s − 2·17-s − 4·19-s − 23-s + 25-s + 8·31-s − 2·35-s + 4·37-s − 4·41-s + 6·43-s − 3·49-s − 2·53-s − 6·59-s − 6·61-s − 2·65-s + 10·67-s + 6·71-s − 14·73-s + 4·79-s + 4·83-s − 2·85-s + 14·89-s + 4·91-s − 4·95-s + 16·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 1.43·31-s − 0.338·35-s + 0.657·37-s − 0.624·41-s + 0.914·43-s − 3/7·49-s − 0.274·53-s − 0.781·59-s − 0.768·61-s − 0.248·65-s + 1.22·67-s + 0.712·71-s − 1.63·73-s + 0.450·79-s + 0.439·83-s − 0.216·85-s + 1.48·89-s + 0.419·91-s − 0.410·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.542848236\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542848236\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.80471005093812083810368469264, −7.01285136000994042163306120700, −6.31425505132561024426837340075, −5.99681280955699235256554095887, −4.89463337080128104812434865075, −4.39897332889049803418476310268, −3.40078877612556301385203162821, −2.62826867634854849007154699351, −1.90093926432267635637948436830, −0.58921553515743088733625076828,
0.58921553515743088733625076828, 1.90093926432267635637948436830, 2.62826867634854849007154699351, 3.40078877612556301385203162821, 4.39897332889049803418476310268, 4.89463337080128104812434865075, 5.99681280955699235256554095887, 6.31425505132561024426837340075, 7.01285136000994042163306120700, 7.80471005093812083810368469264