L(s) = 1 | + (2.23 + 0.00183i)5-s + (−2.45 − 0.988i)7-s + (−2.72 + 1.57i)11-s + (−0.380 + 0.380i)13-s + (1.20 + 0.321i)17-s + (5.59 + 3.22i)19-s + (7.93 − 2.12i)23-s + (4.99 + 0.00820i)25-s + 4.09·29-s + (2.84 + 4.93i)31-s + (−5.48 − 2.21i)35-s + (10.7 − 2.89i)37-s − 7.39i·41-s + (−8.31 + 8.31i)43-s + (−0.420 − 1.56i)47-s + ⋯ |
L(s) = 1 | + (0.999 + 0.000820i)5-s + (−0.927 − 0.373i)7-s + (−0.820 + 0.473i)11-s + (−0.105 + 0.105i)13-s + (0.291 + 0.0780i)17-s + (1.28 + 0.740i)19-s + (1.65 − 0.443i)23-s + (0.999 + 0.00164i)25-s + 0.761·29-s + (0.511 + 0.885i)31-s + (−0.927 − 0.374i)35-s + (1.77 − 0.475i)37-s − 1.15i·41-s + (−1.26 + 1.26i)43-s + (−0.0612 − 0.228i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.805287856\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.805287856\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 - 0.00183i)T \) |
| 7 | \( 1 + (2.45 + 0.988i)T \) |
good | 11 | \( 1 + (2.72 - 1.57i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.380 - 0.380i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.20 - 0.321i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-5.59 - 3.22i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.93 + 2.12i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + (-2.84 - 4.93i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.7 + 2.89i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.39iT - 41T^{2} \) |
| 43 | \( 1 + (8.31 - 8.31i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.420 + 1.56i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.88 + 10.7i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.50 + 9.52i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.74 - 6.47i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 4.89i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 15.1iT - 71T^{2} \) |
| 73 | \( 1 + (-9.39 - 2.51i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.95 + 2.28i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.60 + 2.60i)T + 83iT^{2} \) |
| 89 | \( 1 + (-4.35 + 7.53i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.02 - 2.02i)T + 97iT^{2} \) |
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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
−9.852698100287938138005382729092, −9.115015996826622610378074191405, −8.058774238588006025554974154076, −7.09337667784808872599349781396, −6.46381857780704895194793711491, −5.47489397682763099772177392020, −4.77366570922063427700700830282, −3.33492901780168742805434668827, −2.57285309464503677789769787176, −1.09525921905812743518527300396,
0.980090449991548071862244456724, 2.72848475926599521151117585800, 3.05746103617088487370846623253, 4.78626298377706523914032437087, 5.52374317021138179279023747649, 6.25935969490272547877619312710, 7.10601125213879682729227280451, 8.069674840794042411908927060954, 9.164574249751958405086292905959, 9.550167450922192978921409579276