L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s + 61.0·5-s − 36·6-s + 64·8-s + 81·9-s + 244.·10-s − 36.5·11-s − 144·12-s − 34.5·13-s − 549.·15-s + 256·16-s + 2.06e3·17-s + 324·18-s − 452.·19-s + 977.·20-s − 146.·22-s + 1.68e3·23-s − 576·24-s + 604.·25-s − 138.·26-s − 729·27-s − 4.76e3·29-s − 2.19e3·30-s + 5.26e3·31-s + 1.02e3·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.09·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.772·10-s − 0.0910·11-s − 0.288·12-s − 0.0567·13-s − 0.630·15-s + 0.250·16-s + 1.72·17-s + 0.235·18-s − 0.287·19-s + 0.546·20-s − 0.0643·22-s + 0.663·23-s − 0.204·24-s + 0.193·25-s − 0.0401·26-s − 0.192·27-s − 1.05·29-s − 0.446·30-s + 0.983·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.570386736\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.570386736\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 + 9T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 61.0T + 3.12e3T^{2} \) |
| 11 | \( 1 + 36.5T + 1.61e5T^{2} \) |
| 13 | \( 1 + 34.5T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 452.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.68e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.26e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.28e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.93e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.31e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.27e5T + 8.58e9T^{2} \) |
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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
−10.92983293427933328364130222945, −10.16900144938232334814928508883, −9.299508776084385356618786635867, −7.80461893588277583545641870882, −6.71569989868823558255472362403, −5.71317237477042892511296477286, −5.19021090608180196970982237908, −3.73199070386786848316656292226, −2.33753260515470454110207922308, −1.05593541319359532572723104902,
1.05593541319359532572723104902, 2.33753260515470454110207922308, 3.73199070386786848316656292226, 5.19021090608180196970982237908, 5.71317237477042892511296477286, 6.71569989868823558255472362403, 7.80461893588277583545641870882, 9.299508776084385356618786635867, 10.16900144938232334814928508883, 10.92983293427933328364130222945