L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.258 + 0.965i)3-s + (0.500 + 0.866i)6-s + (1.22 + 1.22i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + 1.73·14-s + 1.00·16-s + (0.707 − 0.707i)17-s + (−0.965 + 0.258i)18-s + (−1.49 + 0.866i)21-s + (−0.866 + 0.5i)24-s + (0.707 − 0.707i)27-s − 1.00i·34-s + (−0.448 + 1.67i)42-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.258 + 0.965i)3-s + (0.500 + 0.866i)6-s + (1.22 + 1.22i)7-s + (0.707 + 0.707i)8-s + (−0.866 − 0.499i)9-s + 1.73·14-s + 1.00·16-s + (0.707 − 0.707i)17-s + (−0.965 + 0.258i)18-s + (−1.49 + 0.866i)21-s + (−0.866 + 0.5i)24-s + (0.707 − 0.707i)27-s − 1.00i·34-s + (−0.448 + 1.67i)42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.013608813\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013608813\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 7 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 59 | \( 1 + 1.73T + T^{2} \) |
| 61 | \( 1 + 2T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−8.982014593825858241837679598397, −8.177768967154453372897805329846, −7.66829594611906290970452464592, −6.28002950630289862149429453782, −5.36100674032071686781412503210, −5.02690751319798241744035848658, −4.33921201271072880209797363795, −3.35122049055856724677830405214, −2.67330172861988962525983764887, −1.69510109899488286754149811894,
1.10574205523218998757335827193, 1.72309836787225465382228339224, 3.27549092526278840409852677593, 4.39315111961188426565824943562, 4.83039341849034140712799160373, 5.83126887461700047123703398221, 6.26008606314280953467150257402, 7.28085424789496293248251234996, 7.59052314148760221478700939798, 8.137465414915758534471404603620