| L(s) = 1 | + 3-s + 2.91·5-s + 7-s + 9-s + 2·11-s − 4.35·13-s + 2.91·15-s + 2.91·17-s − 19-s + 21-s + 2·23-s + 3.51·25-s + 27-s + 7.79·29-s + 6.35·31-s + 2·33-s + 2.91·35-s − 3.83·37-s − 4.35·39-s − 4·41-s + 3.48·43-s + 2.91·45-s − 11.6·47-s + 49-s + 2.91·51-s − 0.918·53-s + 5.83·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.30·5-s + 0.377·7-s + 0.333·9-s + 0.603·11-s − 1.20·13-s + 0.753·15-s + 0.707·17-s − 0.229·19-s + 0.218·21-s + 0.417·23-s + 0.703·25-s + 0.192·27-s + 1.44·29-s + 1.14·31-s + 0.348·33-s + 0.493·35-s − 0.630·37-s − 0.697·39-s − 0.624·41-s + 0.531·43-s + 0.435·45-s − 1.69·47-s + 0.142·49-s + 0.408·51-s − 0.126·53-s + 0.787·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1596 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.861966991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.861966991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.91T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 2.91T + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 + 3.83T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 0.918T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 0.162T + 61T^{2} \) |
| 67 | \( 1 + 5.83T + 67T^{2} \) |
| 71 | \( 1 - 0.918T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−9.547787593558656487925929022361, −8.679409444563745742175813579512, −7.930744521178087974144250441157, −6.93148009464776589357909499628, −6.26099703625189934534839907884, −5.20838821913869724809134731830, −4.54395433180835315932415638654, −3.17037840293012098446784162941, −2.30063876972330465210971497241, −1.31841490372910178337790013612,
1.31841490372910178337790013612, 2.30063876972330465210971497241, 3.17037840293012098446784162941, 4.54395433180835315932415638654, 5.20838821913869724809134731830, 6.26099703625189934534839907884, 6.93148009464776589357909499628, 7.930744521178087974144250441157, 8.679409444563745742175813579512, 9.547787593558656487925929022361
