| L(s) = 1 | − 2.77·2-s + 7.63·3-s − 0.318·4-s − 4.52·5-s − 21.1·6-s + 2.79·7-s + 23.0·8-s + 31.2·9-s + 12.5·10-s − 2.42·12-s + 13·13-s − 7.74·14-s − 34.5·15-s − 61.3·16-s − 56.1·17-s − 86.6·18-s + 40.8·19-s + 1.43·20-s + 21.3·21-s + 83.8·23-s + 175.·24-s − 104.·25-s − 36.0·26-s + 32.5·27-s − 0.888·28-s − 22.9·29-s + 95.6·30-s + ⋯ |
| L(s) = 1 | − 0.979·2-s + 1.46·3-s − 0.0397·4-s − 0.404·5-s − 1.43·6-s + 0.150·7-s + 1.01·8-s + 1.15·9-s + 0.396·10-s − 0.0584·12-s + 0.277·13-s − 0.147·14-s − 0.593·15-s − 0.958·16-s − 0.801·17-s − 1.13·18-s + 0.492·19-s + 0.0160·20-s + 0.221·21-s + 0.760·23-s + 1.49·24-s − 0.836·25-s − 0.271·26-s + 0.231·27-s − 0.00599·28-s − 0.146·29-s + 0.582·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 + 2.77T + 8T^{2} \) |
| 3 | \( 1 - 7.63T + 27T^{2} \) |
| 5 | \( 1 + 4.52T + 125T^{2} \) |
| 7 | \( 1 - 2.79T + 343T^{2} \) |
| 17 | \( 1 + 56.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 40.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 83.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 22.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 199.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 47.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 50.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 330.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 307.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 78.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 430.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 393.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 379.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 87.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 787.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 597.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 998.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.59e3T + 9.12e5T^{2} \) |
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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
−8.718071635355882900244140219576, −7.983397584604677648989268989095, −7.59611008739686331198224413480, −6.67147147900020287003577710166, −5.19731516558116900956437778577, −4.17291806623178574275228152023, −3.44902671963848687246519170969, −2.29692930400407656476673914861, −1.36907691423298497580830275081, 0,
1.36907691423298497580830275081, 2.29692930400407656476673914861, 3.44902671963848687246519170969, 4.17291806623178574275228152023, 5.19731516558116900956437778577, 6.67147147900020287003577710166, 7.59611008739686331198224413480, 7.983397584604677648989268989095, 8.718071635355882900244140219576
