| L(s) = 1 | − 3.67·3-s − 21.0·7-s − 13.4·9-s + 14.9·11-s − 18.5·13-s + 42.5·17-s − 10.1·19-s + 77.3·21-s − 23·23-s + 148.·27-s − 225.·29-s − 259.·31-s − 55.0·33-s + 269.·37-s + 68.2·39-s − 280.·41-s − 327.·43-s + 70.3·47-s + 99.2·49-s − 156.·51-s − 261.·53-s + 37.3·57-s − 26.0·59-s − 787.·61-s + 283.·63-s − 113.·67-s + 84.5·69-s + ⋯ |
| L(s) = 1 | − 0.707·3-s − 1.13·7-s − 0.499·9-s + 0.410·11-s − 0.395·13-s + 0.607·17-s − 0.122·19-s + 0.803·21-s − 0.208·23-s + 1.06·27-s − 1.44·29-s − 1.50·31-s − 0.290·33-s + 1.19·37-s + 0.280·39-s − 1.06·41-s − 1.16·43-s + 0.218·47-s + 0.289·49-s − 0.429·51-s − 0.676·53-s + 0.0867·57-s − 0.0573·59-s − 1.65·61-s + 0.566·63-s − 0.207·67-s + 0.147·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4664311127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4664311127\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 23T \) |
| good | 3 | \( 1 + 3.67T + 27T^{2} \) |
| 7 | \( 1 + 21.0T + 343T^{2} \) |
| 11 | \( 1 - 14.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 269.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 280.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 70.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 261.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 26.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 787.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 113.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 11.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 778.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 7.08T + 4.93e5T^{2} \) |
| 83 | \( 1 + 337.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 423.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.87e3T + 9.12e5T^{2} \) |
| 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
−8.819758876472279736752872149036, −7.75481610958158422981825286339, −7.00411626239130761594244026149, −6.17224939843412654891657070407, −5.70733952294231211804440011796, −4.79784347915118068063876923940, −3.66069120339651453883106399516, −2.99161141910904829120712867707, −1.70058979147859934525371517111, −0.30854978076661407237272456388,
0.30854978076661407237272456388, 1.70058979147859934525371517111, 2.99161141910904829120712867707, 3.66069120339651453883106399516, 4.79784347915118068063876923940, 5.70733952294231211804440011796, 6.17224939843412654891657070407, 7.00411626239130761594244026149, 7.75481610958158422981825286339, 8.819758876472279736752872149036
