| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 3·11-s + 6·13-s + 15-s + 3·17-s + 19-s − 3·21-s + 4·23-s − 4·25-s − 27-s + 10·29-s + 2·31-s − 3·33-s − 3·35-s − 8·37-s − 6·39-s − 8·41-s + 43-s − 45-s + 3·47-s + 2·49-s − 3·51-s + 6·53-s − 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.66·13-s + 0.258·15-s + 0.727·17-s + 0.229·19-s − 0.654·21-s + 0.834·23-s − 4/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s − 0.522·33-s − 0.507·35-s − 1.31·37-s − 0.960·39-s − 1.24·41-s + 0.152·43-s − 0.149·45-s + 0.437·47-s + 2/7·49-s − 0.420·51-s + 0.824·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.073624547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.073624547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 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
−8.487833236481180412785381133395, −7.897976212569257829281969532590, −6.98409637171743183069069898742, −6.32238684276684255747348589201, −5.51580677194910377518131764932, −4.74202626042496205996701596375, −3.98122783789502051053139017405, −3.20617720726784558453701985974, −1.61659551516868209509642661637, −0.989708895420195378653792366927,
0.989708895420195378653792366927, 1.61659551516868209509642661637, 3.20617720726784558453701985974, 3.98122783789502051053139017405, 4.74202626042496205996701596375, 5.51580677194910377518131764932, 6.32238684276684255747348589201, 6.98409637171743183069069898742, 7.897976212569257829281969532590, 8.487833236481180412785381133395
