L(s) = 1 | + 3-s + 3·5-s − 7-s + 9-s − 4·11-s + 3·13-s + 3·15-s − 21-s + 23-s + 4·25-s + 27-s + 29-s + 2·31-s − 4·33-s − 3·35-s − 5·37-s + 3·39-s + 5·41-s + 7·43-s + 3·45-s + 3·47-s + 49-s + 12·53-s − 12·55-s + 2·59-s − 6·61-s − 63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.832·13-s + 0.774·15-s − 0.218·21-s + 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.185·29-s + 0.359·31-s − 0.696·33-s − 0.507·35-s − 0.821·37-s + 0.480·39-s + 0.780·41-s + 1.06·43-s + 0.447·45-s + 0.437·47-s + 1/7·49-s + 1.64·53-s − 1.61·55-s + 0.260·59-s − 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.251148097\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.251148097\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−7.85888997955302046361976834841, −7.21812822048976201667847480244, −6.37509251121743895517907328068, −5.79258289508926464052999400354, −5.22396927104121919692065080765, −4.28274740796210790353117298451, −3.33942564937817058982597819231, −2.58190341384110815345886965201, −1.99851285834170224838718416988, −0.881000111823031885943165753469,
0.881000111823031885943165753469, 1.99851285834170224838718416988, 2.58190341384110815345886965201, 3.33942564937817058982597819231, 4.28274740796210790353117298451, 5.22396927104121919692065080765, 5.79258289508926464052999400354, 6.37509251121743895517907328068, 7.21812822048976201667847480244, 7.85888997955302046361976834841