L(s) = 1 | + (−0.258 − 0.965i)3-s + (1.15 − 2.38i)7-s + (−0.866 + 0.499i)9-s + (−1.36 + 2.36i)11-s + (0.707 − 0.707i)13-s + (−6.50 + 1.74i)17-s + (4.23 + 7.33i)19-s + (−2.59 − 0.500i)21-s + (−1.93 + 7.20i)23-s + (0.707 + 0.707i)27-s + 6.92i·29-s + (−1.73 − i)31-s + (2.63 + 0.707i)33-s + (10.8 + 2.89i)37-s + (−0.866 − 0.500i)39-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.557i)3-s + (0.436 − 0.899i)7-s + (−0.288 + 0.166i)9-s + (−0.411 + 0.713i)11-s + (0.196 − 0.196i)13-s + (−1.57 + 0.422i)17-s + (0.970 + 1.68i)19-s + (−0.566 − 0.109i)21-s + (−0.402 + 1.50i)23-s + (0.136 + 0.136i)27-s + 1.28i·29-s + (−0.311 − 0.179i)31-s + (0.459 + 0.123i)33-s + (1.77 + 0.476i)37-s + (−0.138 − 0.0800i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.268396300\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.268396300\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.15 + 2.38i)T \) |
good | 11 | \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T - 13iT^{2} \) |
| 17 | \( 1 + (6.50 - 1.74i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.23 - 7.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.93 - 7.20i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + (1.73 + i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.8 - 2.89i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + 4.19iT - 41T^{2} \) |
| 43 | \( 1 + (-0.378 - 0.378i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.63 + 9.84i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-8.43 + 2.26i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (4.63 - 8.02i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (8.42 - 4.86i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.81 + 6.76i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (-3.98 - 14.8i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.96 + 1.13i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.34 + 3.34i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.63 - 6.29i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.50 + 2.50i)T + 97iT^{2} \) |
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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
−9.175318749046226839816693359712, −8.238287738651207499520140637247, −7.52802484877049069021398364683, −7.11766636690567484754445882377, −6.05302249116402456234356680562, −5.30372399129906014221837181770, −4.29743339738433698692590155124, −3.49671590572088552863426183028, −2.09253516839301466508510639566, −1.24383611088961721814506272273,
0.48119969604293561772058938689, 2.35713416241932351311526819264, 2.91159804583901531889768829041, 4.43053434019972084412731064860, 4.76756816378796416550097188146, 5.88412693812423641964924927699, 6.40697268141724753936572443033, 7.55336511486862815021908094704, 8.400076743636394089902038205368, 9.099648613994807345454490992897