L(s) = 1 | − 0.363·3-s − 2.86·9-s + 5.14·11-s + 4.64·13-s − 3.86·17-s + 0.778·19-s + 5.00·23-s + 2.13·27-s + 9.42·29-s − 4.72·31-s − 1.86·33-s + 6·37-s − 1.68·39-s + 1.00·41-s − 7.00·43-s + 11.4·47-s + 1.40·51-s + 7.55·53-s − 0.282·57-s − 12.5·59-s − 11.5·61-s − 11.7·67-s − 1.82·69-s + 2.72·71-s − 5.00·73-s + 5.68·79-s + 7.82·81-s + ⋯ |
L(s) = 1 | − 0.209·3-s − 0.955·9-s + 1.55·11-s + 1.28·13-s − 0.938·17-s + 0.178·19-s + 1.04·23-s + 0.410·27-s + 1.74·29-s − 0.848·31-s − 0.325·33-s + 0.986·37-s − 0.270·39-s + 0.157·41-s − 1.06·43-s + 1.66·47-s + 0.196·51-s + 1.03·53-s − 0.0374·57-s − 1.62·59-s − 1.47·61-s − 1.43·67-s − 0.219·69-s + 0.323·71-s − 0.586·73-s + 0.639·79-s + 0.869·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.161887189\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.161887189\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.363T + 3T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 3.86T + 17T^{2} \) |
| 19 | \( 1 - 0.778T + 19T^{2} \) |
| 23 | \( 1 - 5.00T + 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 + 4.72T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 1.00T + 41T^{2} \) |
| 43 | \( 1 + 7.00T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 7.55T + 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + 5.00T + 73T^{2} \) |
| 79 | \( 1 - 5.68T + 79T^{2} \) |
| 83 | \( 1 - 4.67T + 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 - 1.58T + 97T^{2} \) |
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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
−7.64831320827804882756454769487, −6.79105290325709672229702434486, −6.28232435994641961659356757374, −5.85614006335336605792913412113, −4.84452767398691984449136115788, −4.21004330862728433159948885309, −3.41456027361489309763597082002, −2.71711501961668166795343912599, −1.54986673847021639396573650215, −0.75552357921717616885710799060,
0.75552357921717616885710799060, 1.54986673847021639396573650215, 2.71711501961668166795343912599, 3.41456027361489309763597082002, 4.21004330862728433159948885309, 4.84452767398691984449136115788, 5.85614006335336605792913412113, 6.28232435994641961659356757374, 6.79105290325709672229702434486, 7.64831320827804882756454769487