L(s) = 1 | + (−1.30 − 1.14i)3-s + (−0.164 + 0.284i)5-s + (0.5 + 0.866i)7-s + (0.400 + 2.97i)9-s + (0.664 + 1.15i)11-s + (−1.53 + 2.66i)13-s + (0.539 − 0.183i)15-s + 7.35·17-s − 2.93·19-s + (0.335 − 1.69i)21-s + (3.34 − 5.79i)23-s + (2.44 + 4.23i)25-s + (2.86 − 4.33i)27-s + (3.88 + 6.72i)29-s + (1.63 − 2.83i)31-s + ⋯ |
L(s) = 1 | + (−0.752 − 0.658i)3-s + (−0.0735 + 0.127i)5-s + (0.188 + 0.327i)7-s + (0.133 + 0.991i)9-s + (0.200 + 0.347i)11-s + (−0.426 + 0.739i)13-s + (0.139 − 0.0474i)15-s + 1.78·17-s − 0.673·19-s + (0.0732 − 0.370i)21-s + (0.697 − 1.20i)23-s + (0.489 + 0.847i)25-s + (0.552 − 0.833i)27-s + (0.720 + 1.24i)29-s + (0.293 − 0.508i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09820 + 0.170627i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09820 + 0.170627i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.30 + 1.14i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.164 - 0.284i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.664 - 1.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.53 - 2.66i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 7.35T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + (-3.34 + 5.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.88 - 6.72i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.63 + 2.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.329T + 37T^{2} \) |
| 41 | \( 1 + (0.135 - 0.234i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.48 - 9.49i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.571 + 0.989i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6.42T + 53T^{2} \) |
| 59 | \( 1 + (0.372 - 0.644i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.42 + 7.65i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.28 - 7.42i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.60T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + (-0.628 - 1.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.0316 - 0.0548i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + (-5.51 - 9.55i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.02550704662745611441903363053, −10.27910229337915099364377496312, −9.183252685593243185611899636260, −8.109527719335923602726685289555, −7.19916472461921078201396894742, −6.44327077885485953208042260796, −5.37850851240329298770211146793, −4.48478535740684108563758304794, −2.77362272530837118889519092522, −1.32209211604700397364856339217,
0.876628592919545967116490300346, 3.11389040894688474593759631171, 4.21203916665755600646459862584, 5.26462876744088257218187087337, 6.00752989285024510795109340379, 7.21274867438789936318476390698, 8.188544555754332579208854213581, 9.279903581178897556857343105548, 10.24450132917148946864443081427, 10.64125613526712368171223513946