| L(s) = 1 | + 2-s − 3-s + 4-s + 0.856·5-s − 6-s + 8-s + 9-s + 0.856·10-s + 3.13·11-s − 12-s − 2.89·13-s − 0.856·15-s + 16-s + 2.29·17-s + 18-s − 19-s + 0.856·20-s + 3.13·22-s + 5.78·23-s − 24-s − 4.26·25-s − 2.89·26-s − 27-s + 6.14·29-s − 0.856·30-s − 3.04·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.382·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.270·10-s + 0.945·11-s − 0.288·12-s − 0.802·13-s − 0.221·15-s + 0.250·16-s + 0.556·17-s + 0.235·18-s − 0.229·19-s + 0.191·20-s + 0.668·22-s + 1.20·23-s − 0.204·24-s − 0.853·25-s − 0.567·26-s − 0.192·27-s + 1.14·29-s − 0.156·30-s − 0.547·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.028989348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.028989348\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 0.856T + 5T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 23 | \( 1 - 5.78T + 23T^{2} \) |
| 29 | \( 1 - 6.14T + 29T^{2} \) |
| 31 | \( 1 + 3.04T + 31T^{2} \) |
| 37 | \( 1 - 5.86T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 - 9.78T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 4.05T + 71T^{2} \) |
| 73 | \( 1 - 5.54T + 73T^{2} \) |
| 79 | \( 1 + 4.88T + 79T^{2} \) |
| 83 | \( 1 - 5.86T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
| 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
−7.964416216387960421163578968794, −7.15637790640664089358313340836, −6.62231934523127665256981164967, −5.93086844042098607414976955233, −5.25065499874644408232673107284, −4.59012614643605254631979560960, −3.81889030403948948840802183614, −2.89009355460509420352179645553, −1.92375937773845908669945090819, −0.888020471345344422688321449838,
0.888020471345344422688321449838, 1.92375937773845908669945090819, 2.89009355460509420352179645553, 3.81889030403948948840802183614, 4.59012614643605254631979560960, 5.25065499874644408232673107284, 5.93086844042098607414976955233, 6.62231934523127665256981164967, 7.15637790640664089358313340836, 7.964416216387960421163578968794
