L(s) = 1 | + 4-s − 5-s + 9-s + 16-s − 17-s − 19-s − 20-s − 31-s + 36-s − 43-s − 45-s + 49-s − 53-s + 64-s − 67-s − 68-s − 71-s − 76-s − 79-s − 80-s + 81-s + 2·83-s + 85-s + 2·89-s + 95-s + 2·103-s − 109-s + ⋯ |
L(s) = 1 | + 4-s − 5-s + 9-s + 16-s − 17-s − 19-s − 20-s − 31-s + 36-s − 43-s − 45-s + 49-s − 53-s + 64-s − 67-s − 68-s − 71-s − 76-s − 79-s − 80-s + 81-s + 2·83-s + 85-s + 2·89-s + 95-s + 2·103-s − 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8602296346\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8602296346\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 331 | \( 1 - T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−11.74292826314796643656111275461, −10.95237445072461570525508844139, −10.23266083055183576207708023595, −8.903706172792838095372521708606, −7.79180940520932362394377395805, −7.10385284471483772626094148120, −6.20835629064935385794746843377, −4.60474706744117454383959704076, −3.57222521465823825976349946309, −1.98816870243366142600134302603,
1.98816870243366142600134302603, 3.57222521465823825976349946309, 4.60474706744117454383959704076, 6.20835629064935385794746843377, 7.10385284471483772626094148120, 7.79180940520932362394377395805, 8.903706172792838095372521708606, 10.23266083055183576207708023595, 10.95237445072461570525508844139, 11.74292826314796643656111275461