L(s) = 1 | + (0.923 − 0.382i)2-s + (0.707 − 0.707i)3-s + (0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (0.382 − 0.923i)6-s + (0.382 − 0.923i)8-s − 1.00i·9-s + (0.707 + 0.707i)10-s − 1.84·11-s − i·12-s + (0.707 + 0.707i)13-s + (0.923 + 0.382i)15-s − i·16-s + (−0.382 − 0.923i)18-s + (0.923 + 0.382i)20-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)2-s + (0.707 − 0.707i)3-s + (0.707 − 0.707i)4-s + (0.382 + 0.923i)5-s + (0.382 − 0.923i)6-s + (0.382 − 0.923i)8-s − 1.00i·9-s + (0.707 + 0.707i)10-s − 1.84·11-s − i·12-s + (0.707 + 0.707i)13-s + (0.923 + 0.382i)15-s − i·16-s + (−0.382 − 0.923i)18-s + (0.923 + 0.382i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.373876572\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.373876572\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 + 0.382i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.382 - 0.923i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.84T + T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - 0.765T + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 0.765iT - T^{2} \) |
| 61 | \( 1 - 1.41iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.84iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + (-1.30 + 1.30i)T - iT^{2} \) |
| 89 | \( 1 + 1.84iT - T^{2} \) |
| 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
−9.765833886599392280571018462028, −8.622798402181741786133177316647, −7.65325404359676388504374366014, −7.03470007496358397332716907787, −6.20965814724593940114544523626, −5.52396096755768833074596797650, −4.28673304593837246257309732499, −3.17222457461606376476687522728, −2.62207775017757905383560083936, −1.67722487981037798104266740889,
2.06177190652976047698193965960, 2.99144432707594558910151211399, 3.90365071498691143984784925395, 5.05698651229284038333836895809, 5.21460546762718955963022131618, 6.23826340579759278980009614942, 7.65155648192063212717038828520, 8.084809603351847383504296666306, 8.730911112421700855270147783204, 9.763673799756293318120286833419