| L(s) = 1 | − 3-s − 3·5-s + 7-s + 9-s + 6·11-s − 13-s + 3·15-s − 8·17-s − 19-s − 21-s + 23-s + 4·25-s − 27-s − 5·29-s + 3·31-s − 6·33-s − 3·35-s + 12·37-s + 39-s + 10·41-s − 11·43-s − 3·45-s − 3·47-s + 49-s + 8·51-s − 53-s − 18·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.277·13-s + 0.774·15-s − 1.94·17-s − 0.229·19-s − 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s − 0.928·29-s + 0.538·31-s − 1.04·33-s − 0.507·35-s + 1.97·37-s + 0.160·39-s + 1.56·41-s − 1.67·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s + 1.12·51-s − 0.137·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.041358866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.041358866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−9.053764363231988019282930388942, −8.251806928623370947626015876665, −7.46860265227845663427645642544, −6.68026369156152162166263035953, −6.16119822235438323166338829652, −4.71384218644575507810438694512, −4.32745424761876713888572615669, −3.55544519955635390743990765567, −2.05262368051328551108577962292, −0.68821949034666651341211878873,
0.68821949034666651341211878873, 2.05262368051328551108577962292, 3.55544519955635390743990765567, 4.32745424761876713888572615669, 4.71384218644575507810438694512, 6.16119822235438323166338829652, 6.68026369156152162166263035953, 7.46860265227845663427645642544, 8.251806928623370947626015876665, 9.053764363231988019282930388942
