| L(s) = 1 | − 5-s + 7-s + 2·13-s − 2·17-s + 25-s − 6·29-s − 4·31-s − 35-s − 2·37-s − 10·41-s − 4·43-s + 49-s − 2·53-s + 4·59-s + 6·61-s − 2·65-s + 12·67-s + 12·71-s − 2·73-s + 4·83-s + 2·85-s − 10·89-s + 2·91-s − 2·97-s − 6·101-s − 8·103-s − 12·107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.554·13-s − 0.485·17-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.169·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1/7·49-s − 0.274·53-s + 0.520·59-s + 0.768·61-s − 0.248·65-s + 1.46·67-s + 1.42·71-s − 0.234·73-s + 0.439·83-s + 0.216·85-s − 1.05·89-s + 0.209·91-s − 0.203·97-s − 0.597·101-s − 0.788·103-s − 1.16·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.047644687937486999802796461932, −7.07220556070134719930712859938, −6.62373391470830201366455452089, −5.55560525999466039347672876984, −5.02188252261114282661809471903, −3.99515848614807672768134354832, −3.50138285925466710968861847297, −2.32535474697970751879645148778, −1.39297831286557829467848962739, 0,
1.39297831286557829467848962739, 2.32535474697970751879645148778, 3.50138285925466710968861847297, 3.99515848614807672768134354832, 5.02188252261114282661809471903, 5.55560525999466039347672876984, 6.62373391470830201366455452089, 7.07220556070134719930712859938, 8.047644687937486999802796461932
