L(s) = 1 | − 9·3-s + 93.5·5-s − 209.·7-s + 81·9-s + 636.·11-s + 505.·13-s − 842.·15-s − 2.15e3·17-s + 2.32e3·19-s + 1.88e3·21-s − 529·23-s + 5.63e3·25-s − 729·27-s + 4.69e3·29-s + 6.75e3·31-s − 5.73e3·33-s − 1.95e4·35-s − 4.26e3·37-s − 4.54e3·39-s − 7.06e3·41-s − 1.70e4·43-s + 7.57e3·45-s + 7.92e3·47-s + 2.69e4·49-s + 1.94e4·51-s + 8.67e3·53-s + 5.95e4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.67·5-s − 1.61·7-s + 0.333·9-s + 1.58·11-s + 0.828·13-s − 0.966·15-s − 1.81·17-s + 1.47·19-s + 0.931·21-s − 0.208·23-s + 1.80·25-s − 0.192·27-s + 1.03·29-s + 1.26·31-s − 0.915·33-s − 2.70·35-s − 0.512·37-s − 0.478·39-s − 0.656·41-s − 1.40·43-s + 0.557·45-s + 0.523·47-s + 1.60·49-s + 1.04·51-s + 0.424·53-s + 2.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.576400644\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.576400644\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
good | 5 | \( 1 - 93.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 209.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 636.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 505.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.15e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.32e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.75e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.26e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.06e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.70e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.92e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 8.67e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.16e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.78e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.36e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.63e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.57e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.08e3T + 8.58e9T^{2} \) |
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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
−9.231244610549003154031414290796, −8.751028218845816595423573545942, −6.84671613555019933522412679346, −6.48917063422468663541595790549, −6.08291169017388298414325666475, −5.01121497443678129830341853916, −3.82805973519372387558647605339, −2.81666033484278120716602696254, −1.63921453307536356823721911495, −0.73921905454901733583135987464,
0.73921905454901733583135987464, 1.63921453307536356823721911495, 2.81666033484278120716602696254, 3.82805973519372387558647605339, 5.01121497443678129830341853916, 6.08291169017388298414325666475, 6.48917063422468663541595790549, 6.84671613555019933522412679346, 8.751028218845816595423573545942, 9.231244610549003154031414290796