| L(s) = 1 | − i·2-s − 0.732i·3-s − 4-s − 0.732·6-s − i·7-s + i·8-s + 2.46·9-s + 11-s + 0.732i·12-s + 5.46i·13-s − 14-s + 16-s − 3.46i·17-s − 2.46i·18-s − 3.26·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.422i·3-s − 0.5·4-s − 0.298·6-s − 0.377i·7-s + 0.353i·8-s + 0.821·9-s + 0.301·11-s + 0.211i·12-s + 1.51i·13-s − 0.267·14-s + 0.250·16-s − 0.840i·17-s − 0.580i·18-s − 0.749·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.919225781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.919225781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 0.732iT - 3T^{2} \) |
| 13 | \( 1 - 5.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 - 2.19iT - 23T^{2} \) |
| 29 | \( 1 - 1.26T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 2.73iT - 37T^{2} \) |
| 41 | \( 1 - 8.19T + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 + 10.7iT - 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 - 6.39iT - 73T^{2} \) |
| 79 | \( 1 - 1.80T + 79T^{2} \) |
| 83 | \( 1 - 4.39iT - 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 16.5iT - 97T^{2} \) |
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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
−8.420269925523828982531408119555, −7.60294514490307719085641529587, −6.84006966315336484608424711259, −6.38540565282758730202714169568, −5.12706148188751085476362854655, −4.31900031461649680438315607664, −3.86065932318026988272113766546, −2.59252605085204713946045447605, −1.77716866454394280728398949340, −0.865374769396671347688845271207,
0.792041507876167824447267010982, 2.19029512059966163251576509320, 3.38668970942343803645993342607, 4.15193840301951543148447999558, 4.88535195895370744197774625630, 5.73855956595184080468036570108, 6.29631056636058699015013505278, 7.18984796341574089602096119853, 7.88203138628537732847063092961, 8.575445103908399854081555647646
