| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 4·13-s + 5·17-s + 19-s − 6·21-s + 2·23-s − 9·27-s − 8·29-s + 10·31-s − 6·37-s + 12·39-s + 3·41-s − 4·43-s − 4·47-s − 3·49-s − 15·51-s + 6·53-s − 3·57-s − 8·59-s + 10·61-s + 12·63-s − 67-s − 6·69-s − 12·71-s + 3·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 1.10·13-s + 1.21·17-s + 0.229·19-s − 1.30·21-s + 0.417·23-s − 1.73·27-s − 1.48·29-s + 1.79·31-s − 0.986·37-s + 1.92·39-s + 0.468·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 2.10·51-s + 0.824·53-s − 0.397·57-s − 1.04·59-s + 1.28·61-s + 1.51·63-s − 0.122·67-s − 0.722·69-s − 1.42·71-s + 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078459092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078459092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.85414132886411, −12.53187733563193, −11.98419765451013, −11.58113955722064, −11.50423210437249, −10.77835099528788, −10.29359497918336, −9.992159190000727, −9.568534769782978, −8.839815914121758, −8.222069779588460, −7.581132895554293, −7.347287371833091, −6.815717067461508, −6.147825617360329, −5.794475752226039, −5.175401754116664, −4.907569761760039, −4.538837628780275, −3.744586736298942, −3.144238061221675, −2.292145315204809, −1.606097296547197, −1.057438757515623, −0.3813280301358895,
0.3813280301358895, 1.057438757515623, 1.606097296547197, 2.292145315204809, 3.144238061221675, 3.744586736298942, 4.538837628780275, 4.907569761760039, 5.175401754116664, 5.794475752226039, 6.147825617360329, 6.815717067461508, 7.347287371833091, 7.581132895554293, 8.222069779588460, 8.839815914121758, 9.568534769782978, 9.992159190000727, 10.29359497918336, 10.77835099528788, 11.50423210437249, 11.58113955722064, 11.98419765451013, 12.53187733563193, 12.85414132886411
