L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + 1.00·6-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (0.707 + 0.707i)12-s − 1.00·16-s + (−1.41 − 1.41i)17-s + (0.707 − 0.707i)18-s + 2i·19-s + 1.00i·24-s + (−0.707 − 0.707i)27-s + (−0.707 − 0.707i)32-s − 2.00i·34-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + 1.00·6-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (0.707 + 0.707i)12-s − 1.00·16-s + (−1.41 − 1.41i)17-s + (0.707 − 0.707i)18-s + 2i·19-s + 1.00i·24-s + (−0.707 − 0.707i)27-s + (−0.707 − 0.707i)32-s − 2.00i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.502874956\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.502874956\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 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
−11.30549611759141979468412011510, −9.833632625981620163055273367562, −8.872937880362455034599809525358, −8.151202467423163573123564759529, −7.29711525702640386291335044664, −6.58737435379548800679241011965, −5.61989522060658386020253946963, −4.35853059717107534012039461718, −3.32480159013121887546173704780, −2.16481294294981664926714320973,
2.05537942047708543105388562806, 3.05179862368931192218855114284, 4.21470765862868866762739786483, 4.80417957660723987035738135581, 6.05759419280903329947891325220, 7.13116825587942048008052427899, 8.586086693821606002183003002037, 9.124935475141901553935517229121, 10.08290943075282580964016159677, 10.92142427206792842570524286891