L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 3.84·5-s − 36·6-s + 56.0·7-s − 64·8-s + 81·9-s + 15.3·10-s + 11.9·11-s + 144·12-s − 332.·13-s − 224.·14-s − 34.6·15-s + 256·16-s − 542.·17-s − 324·18-s − 1.03e3·19-s − 61.5·20-s + 504.·21-s − 47.7·22-s + 2.80e3·23-s − 576·24-s − 3.11e3·25-s + 1.33e3·26-s + 729·27-s + 896.·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0688·5-s − 0.408·6-s + 0.432·7-s − 0.353·8-s + 0.333·9-s + 0.0486·10-s + 0.0297·11-s + 0.288·12-s − 0.546·13-s − 0.305·14-s − 0.0397·15-s + 0.250·16-s − 0.454·17-s − 0.235·18-s − 0.657·19-s − 0.0344·20-s + 0.249·21-s − 0.0210·22-s + 1.10·23-s − 0.204·24-s − 0.995·25-s + 0.386·26-s + 0.192·27-s + 0.216·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
good | 5 | \( 1 + 3.84T + 3.12e3T^{2} \) |
| 7 | \( 1 - 56.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 11.9T + 1.61e5T^{2} \) |
| 13 | \( 1 + 332.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 542.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.43e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.13e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.28e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.93e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.81e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.20e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.37e5T + 8.58e9T^{2} \) |
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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
−10.04580415694351614417350847293, −9.165803481455451176153064459553, −8.386515121789716606118508927583, −7.52158290357035042322686014661, −6.64441718396172687071709854885, −5.24563486241719942591110026727, −3.96394634759925839405450330053, −2.60586968759062582125539098854, −1.56068977163600938526275351376, 0,
1.56068977163600938526275351376, 2.60586968759062582125539098854, 3.96394634759925839405450330053, 5.24563486241719942591110026727, 6.64441718396172687071709854885, 7.52158290357035042322686014661, 8.386515121789716606118508927583, 9.165803481455451176153064459553, 10.04580415694351614417350847293