| L(s) = 1 | − 2·2-s + 5.52·3-s + 4·4-s − 11.0·6-s + 26.7·7-s − 8·8-s + 3.51·9-s + 12.5·11-s + 22.0·12-s + 15.1·13-s − 53.4·14-s + 16·16-s + 107.·17-s − 7.02·18-s − 26.9·19-s + 147.·21-s − 25.1·22-s + 23·23-s − 44.1·24-s − 30.2·26-s − 129.·27-s + 106.·28-s + 88.1·29-s − 128.·31-s − 32·32-s + 69.5·33-s − 215.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.06·3-s + 0.5·4-s − 0.751·6-s + 1.44·7-s − 0.353·8-s + 0.130·9-s + 0.344·11-s + 0.531·12-s + 0.322·13-s − 1.02·14-s + 0.250·16-s + 1.53·17-s − 0.0919·18-s − 0.325·19-s + 1.53·21-s − 0.243·22-s + 0.208·23-s − 0.375·24-s − 0.227·26-s − 0.924·27-s + 0.721·28-s + 0.564·29-s − 0.742·31-s − 0.176·32-s + 0.366·33-s − 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.972089859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.972089859\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 5.52T + 27T^{2} \) |
| 7 | \( 1 - 26.7T + 343T^{2} \) |
| 11 | \( 1 - 12.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.9T + 6.85e3T^{2} \) |
| 29 | \( 1 - 88.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 424.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 3.45T + 1.03e5T^{2} \) |
| 53 | \( 1 - 79.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 806.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 655.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 127.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 690.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 910.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.53e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 77.2T + 9.12e5T^{2} \) |
| 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
−9.271865556749890620588334387538, −8.481549868362664115384380994581, −7.981359198133026767323307550786, −7.43625023423833105472493474616, −6.15748445456029889452394717951, −5.16080486158081574594724919798, −3.98279380269110542819760524663, −2.95746584799320193694242354812, −1.92081932614490225894649457304, −1.01381338494793330894215206603,
1.01381338494793330894215206603, 1.92081932614490225894649457304, 2.95746584799320193694242354812, 3.98279380269110542819760524663, 5.16080486158081574594724919798, 6.15748445456029889452394717951, 7.43625023423833105472493474616, 7.981359198133026767323307550786, 8.481549868362664115384380994581, 9.271865556749890620588334387538
