L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 6·13-s − 2·17-s − 19-s + 4·21-s + 2·23-s + 4·27-s − 2·29-s + 4·31-s + 10·37-s + 12·39-s − 10·41-s − 6·43-s + 6·47-s − 3·49-s + 4·51-s − 6·53-s + 2·57-s − 4·59-s + 2·61-s − 2·63-s + 2·67-s − 4·69-s + 12·71-s + 6·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 1.66·13-s − 0.485·17-s − 0.229·19-s + 0.872·21-s + 0.417·23-s + 0.769·27-s − 0.371·29-s + 0.718·31-s + 1.64·37-s + 1.92·39-s − 1.56·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s + 0.560·51-s − 0.824·53-s + 0.264·57-s − 0.520·59-s + 0.256·61-s − 0.251·63-s + 0.244·67-s − 0.481·69-s + 1.42·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5846787558\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5846787558\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−9.490789073810240372921768117465, −8.428050902816900678520790091659, −7.44252419642717732006965857891, −6.65222207978962864193738585191, −6.14197739847562876784432920368, −5.09815876670602043980308876473, −4.63011150914358404047040649023, −3.28669462002852311257274446715, −2.25348597492360112441016710296, −0.51765685678695547379208238339,
0.51765685678695547379208238339, 2.25348597492360112441016710296, 3.28669462002852311257274446715, 4.63011150914358404047040649023, 5.09815876670602043980308876473, 6.14197739847562876784432920368, 6.65222207978962864193738585191, 7.44252419642717732006965857891, 8.428050902816900678520790091659, 9.490789073810240372921768117465