| L(s) = 1 | − 3-s − 4-s − 5-s + 9-s + 12-s + 15-s + 16-s + 2·17-s + 20-s + 25-s − 27-s − 36-s − 45-s + 2·47-s − 48-s − 2·51-s − 60-s − 64-s − 2·68-s − 75-s − 2·79-s − 80-s + 81-s + 2·83-s − 2·85-s − 100-s + 108-s + ⋯ |
| L(s) = 1 | − 3-s − 4-s − 5-s + 9-s + 12-s + 15-s + 16-s + 2·17-s + 20-s + 25-s − 27-s − 36-s − 45-s + 2·47-s − 48-s − 2·51-s − 60-s − 64-s − 2·68-s − 75-s − 2·79-s − 80-s + 81-s + 2·83-s − 2·85-s − 100-s + 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4806046859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4806046859\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.52580377938168362653094949067, −9.909151566583766236344994422328, −8.894342961929773631682695501434, −7.87492428114133162125186398435, −7.28282769319704211159370509358, −5.93377975101203954663759203466, −5.16755391643435840110863762454, −4.25890350156268443852679853146, −3.40790009797322116825250087996, −0.946824080689985765932164937186,
0.946824080689985765932164937186, 3.40790009797322116825250087996, 4.25890350156268443852679853146, 5.16755391643435840110863762454, 5.93377975101203954663759203466, 7.28282769319704211159370509358, 7.87492428114133162125186398435, 8.894342961929773631682695501434, 9.909151566583766236344994422328, 10.52580377938168362653094949067
