| L(s) = 1 | − 0.450·2-s − 3·3-s − 7.79·4-s + 1.35·6-s − 0.788·7-s + 7.11·8-s + 9·9-s + 67.3·11-s + 23.3·12-s − 28.5·13-s + 0.355·14-s + 59.1·16-s + 10.0·17-s − 4.05·18-s + 149.·19-s + 2.36·21-s − 30.3·22-s + 2.58·23-s − 21.3·24-s + 12.8·26-s − 27·27-s + 6.15·28-s − 29·29-s + 114.·31-s − 83.6·32-s − 202.·33-s − 4.51·34-s + ⋯ |
| L(s) = 1 | − 0.159·2-s − 0.577·3-s − 0.974·4-s + 0.0919·6-s − 0.0425·7-s + 0.314·8-s + 0.333·9-s + 1.84·11-s + 0.562·12-s − 0.608·13-s + 0.00678·14-s + 0.924·16-s + 0.142·17-s − 0.0531·18-s + 1.79·19-s + 0.0245·21-s − 0.294·22-s + 0.0234·23-s − 0.181·24-s + 0.0969·26-s − 0.192·27-s + 0.0415·28-s − 0.185·29-s + 0.664·31-s − 0.461·32-s − 1.06·33-s − 0.0227·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.616363745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.616363745\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 0.450T + 8T^{2} \) |
| 7 | \( 1 + 0.788T + 343T^{2} \) |
| 11 | \( 1 - 67.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 10.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 149.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.58T + 1.21e4T^{2} \) |
| 31 | \( 1 - 114.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 399.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 340.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 242.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 376.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 263.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 395.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 541.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 496.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 81.1T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.04e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 398.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 362.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 402.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.016131290186121865618564993472, −7.83807330737388434799698371361, −7.28102594063034339569483808982, −6.22519449546332642274677670019, −5.59197106295520307364695039111, −4.57829112296476850219372112952, −4.07258750668923793812241983742, −2.99697460317596903302210651314, −1.33978268393279870572231352633, −0.71770530559250330228077940596,
0.71770530559250330228077940596, 1.33978268393279870572231352633, 2.99697460317596903302210651314, 4.07258750668923793812241983742, 4.57829112296476850219372112952, 5.59197106295520307364695039111, 6.22519449546332642274677670019, 7.28102594063034339569483808982, 7.83807330737388434799698371361, 9.016131290186121865618564993472
