L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (−0.707 − 0.707i)5-s − i·7-s + (−1.00 + 1.00i)10-s + (−0.707 − 0.707i)11-s + i·13-s − 1.41·14-s − 0.999·16-s + (−0.707 + 0.707i)17-s + (1 − i)19-s + (0.707 + 0.707i)20-s + (−1.00 + 1.00i)22-s + 1.00i·25-s + 1.41·26-s + ⋯ |
L(s) = 1 | − 1.41i·2-s − 1.00·4-s + (−0.707 − 0.707i)5-s − i·7-s + (−1.00 + 1.00i)10-s + (−0.707 − 0.707i)11-s + i·13-s − 1.41·14-s − 0.999·16-s + (−0.707 + 0.707i)17-s + (1 − i)19-s + (0.707 + 0.707i)20-s + (−1.00 + 1.00i)22-s + 1.00i·25-s + 1.41·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7113428581\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7113428581\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 - iT \) |
good | 2 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 + (-1 + i)T - iT^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (1 + i)T + iT^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−9.155553299753020804272522128712, −8.479817485852909443401654106490, −7.47405667097760737642710297983, −6.80077281425422708493434812369, −5.42403926061772187696951895999, −4.36638692038019493248726688942, −3.94167618332493951033016245345, −2.96245342177771874046217349872, −1.72278458366729844328234675086, −0.52846324518906940756117331930,
2.33685078598129499902922578531, 3.24555355172147206812645862899, 4.57909120856820058183067336954, 5.43864154959947282944467900413, 5.92874984167280087492863516643, 7.18217898327931362595832054857, 7.31278896940280515320478139739, 8.249391397593048941380757687603, 8.804197382695310357694372605655, 9.854742386865629009854504018165