| L(s) = 1 | + i·2-s − 4-s − 3.83i·5-s − 3.36i·7-s − i·8-s + 3.83·10-s − 1.99i·11-s + (−2.51 − 2.58i)13-s + 3.36·14-s + 16-s + 2.60·17-s − 1.99i·19-s + 3.83i·20-s + 1.99·22-s − 4.27·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.71i·5-s − 1.27i·7-s − 0.353i·8-s + 1.21·10-s − 0.600i·11-s + (−0.696 − 0.717i)13-s + 0.900·14-s + 0.250·16-s + 0.630·17-s − 0.456i·19-s + 0.857i·20-s + 0.424·22-s − 0.890·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.190519787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.190519787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (2.51 + 2.58i)T \) |
| good | 5 | \( 1 + 3.83iT - 5T^{2} \) |
| 7 | \( 1 + 3.36iT - 7T^{2} \) |
| 11 | \( 1 + 1.99iT - 11T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 19 | \( 1 + 1.99iT - 19T^{2} \) |
| 23 | \( 1 + 4.27T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 5.28iT - 31T^{2} \) |
| 37 | \( 1 - 5.08iT - 37T^{2} \) |
| 41 | \( 1 - 8.24iT - 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.88iT - 47T^{2} \) |
| 53 | \( 1 + 9.16T + 53T^{2} \) |
| 59 | \( 1 - 7.31iT - 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 + 5.66iT - 67T^{2} \) |
| 71 | \( 1 - 1.94iT - 71T^{2} \) |
| 73 | \( 1 + 8.41iT - 73T^{2} \) |
| 79 | \( 1 + 5.95T + 79T^{2} \) |
| 83 | \( 1 + 2.35iT - 83T^{2} \) |
| 89 | \( 1 - 6.28iT - 89T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−8.621695886485046848313139564305, −7.896611983774669317602090619116, −7.59866337234076407833074576118, −6.33103246411323354787368339452, −5.64340106721139443211853879799, −4.59576774958591035036963989489, −4.40798049289430499239316582809, −3.08604624279658789999108504953, −1.21917425835703811352296963072, −0.45649432733441042269146071010,
1.95337521451477007661418597436, 2.53773428881214272055867916248, 3.33473032653655628895978339421, 4.36043898580011664724198639236, 5.47167929675414834916624813161, 6.24200165571104417467026836861, 7.05779663317484968881926794983, 7.81532171679532690697674164320, 8.795282731386557295879727350535, 9.609542476150854611208946558038
