L(s) = 1 | + (0.707 − 0.707i)5-s + (1 − i)7-s + 1.41i·11-s − 1.00i·25-s − 1.41·29-s − 1.41i·35-s − i·49-s + (1.00 + 1.00i)55-s − 1.41·59-s + (1 + i)73-s + (1.41 + 1.41i)77-s + (1.41 − 1.41i)83-s + (−1 + i)97-s + 1.41i·101-s + (−1 − i)103-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)5-s + (1 − i)7-s + 1.41i·11-s − 1.00i·25-s − 1.41·29-s − 1.41i·35-s − i·49-s + (1.00 + 1.00i)55-s − 1.41·59-s + (1 + i)73-s + (1.41 + 1.41i)77-s + (1.41 − 1.41i)83-s + (−1 + i)97-s + 1.41i·101-s + (−1 − i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.347599652\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.347599652\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1 + i)T - iT^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 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 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 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
−9.648333228706408484649200535162, −8.975108401236366213536030127888, −7.903032492936605643134446050087, −7.40448494976705683909994527514, −6.42423265312709727222128965060, −5.28060923241097287067246004484, −4.68335828631591790826189762781, −3.89937790894559906117786684628, −2.18961229420917321246268900238, −1.36293816973987177981554515946,
1.67726755341189007465008378223, 2.65384282431495531795497896803, 3.61721760789822778265946992051, 5.03784283417454863002743097183, 5.72530047700319781805796430884, 6.31141017574478206698481934846, 7.45275038459915536116857079506, 8.275422161421833902732116497321, 8.986340453809853578920491250991, 9.679446458796625544659644742024