L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.258 + 0.965i)3-s + (0.500 − 0.866i)6-s + (−0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.258 − 0.965i)13-s + (0.866 + 0.500i)14-s + 1.00·16-s + (−0.258 + 0.965i)17-s + (0.965 + 0.258i)18-s + (−0.866 − 0.5i)19-s + (−0.499 − 0.866i)21-s + (−0.866 − 0.5i)24-s + (−0.500 + 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.258 + 0.965i)3-s + (0.500 − 0.866i)6-s + (−0.965 + 0.258i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (−0.258 − 0.965i)13-s + (0.866 + 0.500i)14-s + 1.00·16-s + (−0.258 + 0.965i)17-s + (0.965 + 0.258i)18-s + (−0.866 − 0.5i)19-s + (−0.499 − 0.866i)21-s + (−0.866 − 0.5i)24-s + (−0.500 + 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1276778588\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1276778588\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.965 - 0.258i)T \) |
good | 2 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.707 - 0.707i)T + iT^{2} \) |
| 53 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 - iT - T^{2} \) |
| 83 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.906057537220594783757527754972, −9.328136589317893312844647121106, −8.687857742345498212085428496068, −8.012424276037981049955787168389, −6.62820299876220975412032235697, −5.75021382421293909511570647798, −5.02626659687288173713339754663, −3.70719740993966932509469020224, −2.99856500217957436810707241741, −1.98491876706737872351358226189,
0.11204262707789757538186926695, 1.96924782360785571842396142702, 3.15398381557730017306994834060, 4.03797993435311022904855569012, 5.63494536132755835831409580040, 6.45662592285328315566848526210, 7.08527550923236829500419893813, 7.45437096590041251792979008151, 8.529684826902436470747356274013, 9.077015504427993223130327199205