L(s) = 1 | + (−1.62 + 0.829i)2-s + (1.37 − 1.89i)4-s + (−0.156 − 0.987i)7-s + (−0.382 + 2.41i)8-s + (0.951 + 0.309i)9-s + (−0.978 − 0.207i)11-s + (1.07 + 1.47i)14-s + (−0.657 − 2.02i)16-s + (−1.80 + 0.285i)18-s + (1.76 − 0.472i)22-s + (1.40 + 1.40i)23-s + (−2.08 − 1.06i)28-s + (0.336 + 0.244i)29-s + (1.02 + 1.02i)32-s + (1.89 − 1.37i)36-s + (−0.803 + 0.127i)37-s + ⋯ |
L(s) = 1 | + (−1.62 + 0.829i)2-s + (1.37 − 1.89i)4-s + (−0.156 − 0.987i)7-s + (−0.382 + 2.41i)8-s + (0.951 + 0.309i)9-s + (−0.978 − 0.207i)11-s + (1.07 + 1.47i)14-s + (−0.657 − 2.02i)16-s + (−1.80 + 0.285i)18-s + (1.76 − 0.472i)22-s + (1.40 + 1.40i)23-s + (−2.08 − 1.06i)28-s + (0.336 + 0.244i)29-s + (1.02 + 1.02i)32-s + (1.89 − 1.37i)36-s + (−0.803 + 0.127i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5302470963\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5302470963\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + (0.156 + 0.987i)T \) |
| 11 | \( 1 + (0.978 + 0.207i)T \) |
good | 2 | \( 1 + (1.62 - 0.829i)T + (0.587 - 0.809i)T^{2} \) |
| 3 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-1.40 - 1.40i)T + iT^{2} \) |
| 29 | \( 1 + (-0.336 - 0.244i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.803 - 0.127i)T + (0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (-1.38 + 1.38i)T - iT^{2} \) |
| 47 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (-1.69 + 0.863i)T + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-1.05 + 1.05i)T - iT^{2} \) |
| 71 | \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.459 + 1.41i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.587 - 0.809i)T^{2} \) |
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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
−9.382189339834658091136843270532, −8.592348527516322842063443095506, −7.69527361821854847408302088907, −7.29602155173602019160746884450, −6.76172813602097646558623156336, −5.63541994376740400165904947286, −4.84825823808432951488886103977, −3.47623012729028992152529864606, −1.97815794982261958103525101979, −0.850110757205564688020683918974,
1.06268504102836310132941725105, 2.38115616524130678520491598047, 2.85810705483976897606516935660, 4.20349125105255805306309869019, 5.38230539346057805687042293862, 6.63167663827107064195097643724, 7.23926568672209776831268387436, 8.124540552816921816814460985440, 8.749964588869803372173093426362, 9.374533118663558554405408902776