L(s) = 1 | + (0.847 + 1.13i)2-s − i·3-s + (−0.562 + 1.91i)4-s − i·5-s + (1.13 − 0.847i)6-s − 1.84·7-s + (−2.64 + 0.991i)8-s − 9-s + (1.13 − 0.847i)10-s + 2.86·11-s + (1.91 + 0.562i)12-s − 4.13·13-s + (−1.56 − 2.09i)14-s − 15-s + (−3.36 − 2.15i)16-s − 6.96i·17-s + ⋯ |
L(s) = 1 | + (0.599 + 0.800i)2-s − 0.577i·3-s + (−0.281 + 0.959i)4-s − 0.447i·5-s + (0.462 − 0.346i)6-s − 0.698·7-s + (−0.936 + 0.350i)8-s − 0.333·9-s + (0.357 − 0.268i)10-s + 0.864·11-s + (0.554 + 0.162i)12-s − 1.14·13-s + (−0.418 − 0.559i)14-s − 0.258·15-s + (−0.842 − 0.539i)16-s − 1.69i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.680614996\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.680614996\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.847 - 1.13i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-1.84 + 4.42i)T \) |
good | 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 6.96iT - 17T^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 29 | \( 1 - 9.51T + 29T^{2} \) |
| 31 | \( 1 - 0.644iT - 31T^{2} \) |
| 37 | \( 1 + 10.0iT - 37T^{2} \) |
| 41 | \( 1 + 6.84T + 41T^{2} \) |
| 43 | \( 1 + 0.868T + 43T^{2} \) |
| 47 | \( 1 + 4.69iT - 47T^{2} \) |
| 53 | \( 1 + 3.26iT - 53T^{2} \) |
| 59 | \( 1 - 12.2iT - 59T^{2} \) |
| 61 | \( 1 + 7.83iT - 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 + 12.6iT - 71T^{2} \) |
| 73 | \( 1 + 0.824T + 73T^{2} \) |
| 79 | \( 1 + 4.81T + 79T^{2} \) |
| 83 | \( 1 + 7.08T + 83T^{2} \) |
| 89 | \( 1 - 3.47iT - 89T^{2} \) |
| 97 | \( 1 - 6.83iT - 97T^{2} \) |
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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
−9.292730430532605781372793444401, −8.627134962061307234959537341438, −7.52386042086328171693815536780, −7.03419291387428368804891741392, −6.36111582546834104651499348489, −5.24418087143957864589064962072, −4.71620138477733581994294980161, −3.39210689550004739182967807163, −2.56295531823794977018829573583, −0.58989987005549253353785201656,
1.41176121068275466840118895356, 2.90123542681489619887785871017, 3.44543380190499298365186703101, 4.42653400264665848616238903081, 5.31056796936024645407213764006, 6.25561291505042849720810072425, 6.92541695844923078166991025465, 8.246926773066381031410959630954, 9.265212987569329246812391878784, 10.03348700878432852335882076118