| L(s) = 1 | − 0.685·2-s − 7.53·4-s − 9.28·5-s − 6.04·7-s + 10.6·8-s + 6.35·10-s − 59.2·11-s + 4.14·14-s + 52.9·16-s + 117.·17-s − 49.2·19-s + 69.9·20-s + 40.5·22-s − 23.1·23-s − 38.8·25-s + 45.5·28-s + 145.·29-s + 94.9·31-s − 121.·32-s − 80.4·34-s + 56.1·35-s + 379.·37-s + 33.7·38-s − 98.7·40-s + 268.·41-s − 23.6·43-s + 445.·44-s + ⋯ |
| L(s) = 1 | − 0.242·2-s − 0.941·4-s − 0.830·5-s − 0.326·7-s + 0.470·8-s + 0.201·10-s − 1.62·11-s + 0.0790·14-s + 0.827·16-s + 1.67·17-s − 0.594·19-s + 0.781·20-s + 0.393·22-s − 0.209·23-s − 0.310·25-s + 0.307·28-s + 0.928·29-s + 0.550·31-s − 0.670·32-s − 0.405·34-s + 0.271·35-s + 1.68·37-s + 0.144·38-s − 0.390·40-s + 1.02·41-s − 0.0839·43-s + 1.52·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.685T + 8T^{2} \) |
| 5 | \( 1 + 9.28T + 125T^{2} \) |
| 7 | \( 1 + 6.04T + 343T^{2} \) |
| 11 | \( 1 + 59.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 49.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 23.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 94.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 379.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 268.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 23.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + 301.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 391.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 510.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 520.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 470.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 466.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 314.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 47.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 310.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 216.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 219.T + 9.12e5T^{2} \) |
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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
−8.450122867210808679336727835213, −7.977816005272345913865454384815, −7.51553913929715486995943560657, −6.14195507123331695479226458092, −5.27306462925693859591692923809, −4.48765178869713528882929866208, −3.58592698440924953432762337575, −2.64975082420325305737567355938, −0.925500302341701506420165629701, 0,
0.925500302341701506420165629701, 2.64975082420325305737567355938, 3.58592698440924953432762337575, 4.48765178869713528882929866208, 5.27306462925693859591692923809, 6.14195507123331695479226458092, 7.51553913929715486995943560657, 7.977816005272345913865454384815, 8.450122867210808679336727835213
