L(s) = 1 | − 9·3-s + 47.0·5-s + 114.·7-s + 81·9-s − 121·11-s + 362.·13-s − 423.·15-s − 1.28e3·17-s − 1.49e3·19-s − 1.03e3·21-s − 3.93e3·23-s − 907.·25-s − 729·27-s + 8.95e3·29-s − 3.65e3·31-s + 1.08e3·33-s + 5.41e3·35-s − 1.36e4·37-s − 3.26e3·39-s + 1.78e4·41-s + 7.21e3·43-s + 3.81e3·45-s − 6.37e3·47-s − 3.58e3·49-s + 1.16e4·51-s − 3.06e4·53-s − 5.69e3·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.842·5-s + 0.886·7-s + 0.333·9-s − 0.301·11-s + 0.594·13-s − 0.486·15-s − 1.08·17-s − 0.947·19-s − 0.512·21-s − 1.55·23-s − 0.290·25-s − 0.192·27-s + 1.97·29-s − 0.682·31-s + 0.174·33-s + 0.747·35-s − 1.64·37-s − 0.343·39-s + 1.65·41-s + 0.594·43-s + 0.280·45-s − 0.421·47-s − 0.213·49-s + 0.624·51-s − 1.49·53-s − 0.253·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 11 | \( 1 + 121T \) |
good | 5 | \( 1 - 47.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 114.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 362.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.28e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.93e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.95e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.65e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.36e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.78e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.21e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.37e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.06e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.55e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.14e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.41e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.49e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.95e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.37e3T + 8.58e9T^{2} \) |
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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
−9.759167571990190685898795462642, −8.681248674058293834251913904818, −7.925017025090021949668366680681, −6.59947312899999231808948703655, −5.97964513316687843105665506475, −4.95735107943409163725812278692, −4.08217189718456294193948034574, −2.33489648609629973071855969553, −1.49700049671868930133441170766, 0,
1.49700049671868930133441170766, 2.33489648609629973071855969553, 4.08217189718456294193948034574, 4.95735107943409163725812278692, 5.97964513316687843105665506475, 6.59947312899999231808948703655, 7.925017025090021949668366680681, 8.681248674058293834251913904818, 9.759167571990190685898795462642