| L(s) = 1 | + 5-s − 7-s + 4·11-s + 6·13-s − 4·17-s − 2·19-s + 8·23-s + 25-s − 6·29-s + 4·31-s − 35-s − 8·37-s + 2·41-s + 6·43-s − 6·47-s + 49-s − 2·53-s + 4·55-s + 4·59-s + 6·65-s + 14·67-s − 2·71-s + 6·73-s − 4·77-s + 8·79-s − 8·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s − 0.970·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s − 1.31·37-s + 0.312·41-s + 0.914·43-s − 0.875·47-s + 1/7·49-s − 0.274·53-s + 0.539·55-s + 0.520·59-s + 0.744·65-s + 1.71·67-s − 0.237·71-s + 0.702·73-s − 0.455·77-s + 0.900·79-s − 0.878·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.182575258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.182575258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.925557090567667143426188183157, −8.445976369585896032011753297769, −7.15356662496065697525447623500, −6.52642446903419247050898537618, −6.02755193173207596037771400784, −4.98184613180721557207894221812, −3.97470750611127582029051537416, −3.31206270428353103021750434411, −2.02733763862439592052964201381, −0.996061989335435497925819768139,
0.996061989335435497925819768139, 2.02733763862439592052964201381, 3.31206270428353103021750434411, 3.97470750611127582029051537416, 4.98184613180721557207894221812, 6.02755193173207596037771400784, 6.52642446903419247050898537618, 7.15356662496065697525447623500, 8.445976369585896032011753297769, 8.925557090567667143426188183157
