L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 53.5·5-s + 36·6-s + 183.·7-s + 64·8-s + 81·9-s − 214.·10-s − 294.·11-s + 144·12-s − 126.·13-s + 733.·14-s − 481.·15-s + 256·16-s + 1.07e3·17-s + 324·18-s + 1.24e3·19-s − 856.·20-s + 1.65e3·21-s − 1.17e3·22-s + 768.·23-s + 576·24-s − 261.·25-s − 504.·26-s + 729·27-s + 2.93e3·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.957·5-s + 0.408·6-s + 1.41·7-s + 0.353·8-s + 0.333·9-s − 0.676·10-s − 0.733·11-s + 0.288·12-s − 0.207·13-s + 1.00·14-s − 0.552·15-s + 0.250·16-s + 0.905·17-s + 0.235·18-s + 0.788·19-s − 0.478·20-s + 0.816·21-s − 0.518·22-s + 0.303·23-s + 0.204·24-s − 0.0836·25-s − 0.146·26-s + 0.192·27-s + 0.707·28-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
\(L(3)\) |
\(\approx\) |
\(4.272873804\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.272873804\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 53.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 183.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 294.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 126.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 768.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.52e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.70e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.04e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 229.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.05e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 4.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.11e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.02e5T + 8.58e9T^{2} \) |
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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
−10.87171680690215092341053704403, −9.851761967791969471314762420683, −8.292171450458016033405468660116, −7.937140155346723984850805038994, −7.05157104418855483619219981417, −5.40332654046541961969539629542, −4.64969113170579763288997038806, −3.59379535453674987388800389280, −2.46526688287087057131875594227, −1.04943086684359696792216215938,
1.04943086684359696792216215938, 2.46526688287087057131875594227, 3.59379535453674987388800389280, 4.64969113170579763288997038806, 5.40332654046541961969539629542, 7.05157104418855483619219981417, 7.937140155346723984850805038994, 8.292171450458016033405468660116, 9.851761967791969471314762420683, 10.87171680690215092341053704403