L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s − 1.00i·6-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 1.00·10-s − 1.41·11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)14-s + 1.00·15-s − 1.00·16-s + (−0.707 − 0.707i)18-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s + (0.707 + 0.707i)5-s − 1.00i·6-s + i·7-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + 1.00·10-s − 1.41·11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)14-s + 1.00·15-s − 1.00·16-s + (−0.707 − 0.707i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.746592442\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.746592442\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 - iT \) |
good | 11 | \( 1 + 1.41T + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−10.27275796556190491283122114687, −9.528030369203851765293387248797, −8.678885595731812906746922218699, −7.62747007341311021654808067599, −6.53929113813073356187731439951, −5.84656096314949708592075285408, −4.94618793059937242988507428077, −3.29726981972386314402310601911, −2.63967026322946297514642594706, −1.86498799025918509090511760598,
2.29174084579632205970616944395, 3.43209531423189077168190502280, 4.50221113700582711880768876478, 5.08946748154639789676236287643, 6.01737394771442801533937004071, 7.34791635546242617611968444033, 7.950892122340426596888592729030, 8.765943432568467482948791656006, 9.670493432451313551590235102483, 10.43973505863654009572186558734