L(s) = 1 | + 0.0210·3-s + (0.567 − 0.412i)5-s + (−0.309 + 0.951i)7-s − 2.99·9-s + (−0.883 − 0.641i)11-s + (0.0516 + 0.158i)13-s + (0.0119 − 0.00866i)15-s + (6.53 + 4.74i)17-s + (0.882 − 2.71i)19-s + (−0.00649 + 0.0199i)21-s + (2.42 + 7.46i)23-s + (−1.39 + 4.28i)25-s − 0.126·27-s + (6.32 − 4.59i)29-s + (2.27 + 1.65i)31-s + ⋯ |
L(s) = 1 | + 0.0121·3-s + (0.253 − 0.184i)5-s + (−0.116 + 0.359i)7-s − 0.999·9-s + (−0.266 − 0.193i)11-s + (0.0143 + 0.0440i)13-s + (0.00307 − 0.00223i)15-s + (1.58 + 1.15i)17-s + (0.202 − 0.622i)19-s + (−0.00141 + 0.00435i)21-s + (0.506 + 1.55i)23-s + (−0.278 + 0.857i)25-s − 0.0242·27-s + (1.17 − 0.852i)29-s + (0.408 + 0.296i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.509973727\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.509973727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 41 | \( 1 + (-5.44 - 3.37i)T \) |
good | 3 | \( 1 - 0.0210T + 3T^{2} \) |
| 5 | \( 1 + (-0.567 + 0.412i)T + (1.54 - 4.75i)T^{2} \) |
| 11 | \( 1 + (0.883 + 0.641i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.0516 - 0.158i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-6.53 - 4.74i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.882 + 2.71i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.42 - 7.46i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-6.32 + 4.59i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.27 - 1.65i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 0.945i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-1.79 - 5.53i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-0.776 - 2.38i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (7.52 - 5.46i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.17 + 12.8i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.82 - 5.60i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-3.24 + 2.35i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (5.46 + 3.97i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 3.45T + 79T^{2} \) |
| 83 | \( 1 + 4.51T + 83T^{2} \) |
| 89 | \( 1 + (3.96 - 12.2i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (0.661 - 0.480i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.694367693271256791051014351585, −9.235278770182104095709971731553, −8.149751964613513259142155391183, −7.70884679933268531013286574263, −6.30564147708027065050757491407, −5.72135826237035640190261845851, −4.94922126045059232280901742069, −3.52360135981265158565406950367, −2.75241917634440811639733050740, −1.26015217928248744077650337846,
0.74731458249350066302563502530, 2.52263684576012764283163347574, 3.25350541805218755852603848693, 4.58316894727307540461230391555, 5.48225895507134081144830636945, 6.29695434190739072921891088225, 7.24129940149072619469235101230, 8.084818769842606209337536124795, 8.822537413228216183506276564063, 9.873848698573414067105160919141