L(s) = 1 | + (0.283 + 0.0270i)2-s + (−0.902 − 0.173i)4-s + (0.580 − 0.814i)7-s + (−0.524 − 0.153i)8-s + (0.415 + 0.909i)9-s + (1.56 − 1.23i)11-s + (0.186 − 0.215i)14-s + (0.708 + 0.283i)16-s + (0.0930 + 0.268i)18-s + (0.476 − 0.306i)22-s + (−1.28 + 0.663i)23-s + (−0.959 + 0.281i)25-s + (−0.665 + 0.634i)28-s + (−0.580 − 1.00i)29-s + (0.678 + 0.349i)32-s + ⋯ |
L(s) = 1 | + (0.283 + 0.0270i)2-s + (−0.902 − 0.173i)4-s + (0.580 − 0.814i)7-s + (−0.524 − 0.153i)8-s + (0.415 + 0.909i)9-s + (1.56 − 1.23i)11-s + (0.186 − 0.215i)14-s + (0.708 + 0.283i)16-s + (0.0930 + 0.268i)18-s + (0.476 − 0.306i)22-s + (−1.28 + 0.663i)23-s + (−0.959 + 0.281i)25-s + (−0.665 + 0.634i)28-s + (−0.580 − 1.00i)29-s + (0.678 + 0.349i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 469 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9064509808\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9064509808\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.580 + 0.814i)T \) |
| 67 | \( 1 + (-0.928 - 0.371i)T \) |
good | 2 | \( 1 + (-0.283 - 0.0270i)T + (0.981 + 0.189i)T^{2} \) |
| 3 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 5 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 1.23i)T + (0.235 - 0.971i)T^{2} \) |
| 13 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 17 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 19 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 23 | \( 1 + (1.28 - 0.663i)T + (0.580 - 0.814i)T^{2} \) |
| 29 | \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.888 - 0.458i)T^{2} \) |
| 37 | \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 43 | \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 53 | \( 1 + (0.0623 - 0.0719i)T + (-0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 61 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 71 | \( 1 + (1.54 + 0.297i)T + (0.928 + 0.371i)T^{2} \) |
| 73 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 79 | \( 1 + (-1.21 - 1.16i)T + (0.0475 + 0.998i)T^{2} \) |
| 83 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 89 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−11.29223442940078912713479379554, −10.21886305262581660527282976704, −9.481297285957696905393546847177, −8.405753836330330170487576558438, −7.73752291002473013278380956492, −6.39399192435951223332079666545, −5.43474621008412698546859852632, −4.27346450950258326583875829168, −3.68345339213178962244386008740, −1.46085677850117055015982965937,
1.86294339093555698853766874905, 3.75611783247860969015481572580, 4.36691549510720037531013439518, 5.56239836358190962707234413009, 6.56988081133560916404349946624, 7.71106929493771235793677964722, 8.943615138745397695959197513444, 9.254850900513371679029003929988, 10.20495450729284244175971091687, 11.70995101494106658238948009635