| L(s) = 1 | − 3.11·3-s + 4.50·7-s + 6.72·9-s − 4.33·11-s + 3.72·13-s − 1.11·17-s − 4.50·19-s − 14.0·21-s − 23-s − 11.6·27-s − 8.23·29-s − 1.72·31-s + 13.5·33-s + 0.781·37-s − 11.6·39-s + 3.90·41-s + 8·43-s − 11.4·47-s + 13.3·49-s + 3.49·51-s + 6·53-s + 14.0·57-s + 2.23·59-s + 3.55·61-s + 30.3·63-s + 2.43·67-s + 3.11·69-s + ⋯ |
| L(s) = 1 | − 1.80·3-s + 1.70·7-s + 2.24·9-s − 1.30·11-s + 1.03·13-s − 0.271·17-s − 1.03·19-s − 3.06·21-s − 0.208·23-s − 2.23·27-s − 1.52·29-s − 0.310·31-s + 2.35·33-s + 0.128·37-s − 1.86·39-s + 0.609·41-s + 1.21·43-s − 1.67·47-s + 1.90·49-s + 0.488·51-s + 0.824·53-s + 1.86·57-s + 0.291·59-s + 0.455·61-s + 3.82·63-s + 0.297·67-s + 0.375·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019820715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019820715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 7 | \( 1 - 4.50T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 - 3.72T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 - 0.781T + 37T^{2} \) |
| 41 | \( 1 - 3.90T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 9.45T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 7.69T + 89T^{2} \) |
| 97 | \( 1 - 0.642T + 97T^{2} \) |
| 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
−7.72425662748401041618429419787, −6.95279817218185206939090462296, −6.15261643595461727031474844747, −5.55022630725042443559253024636, −5.14576243476999446657013864727, −4.44814144080839845998164149001, −3.85332906184975640402054263671, −2.25767281119315646884215174556, −1.57305575735834759362924376693, −0.55469133178485888853076162720,
0.55469133178485888853076162720, 1.57305575735834759362924376693, 2.25767281119315646884215174556, 3.85332906184975640402054263671, 4.44814144080839845998164149001, 5.14576243476999446657013864727, 5.55022630725042443559253024636, 6.15261643595461727031474844747, 6.95279817218185206939090462296, 7.72425662748401041618429419787
