L(s) = 1 | − 0.315i·2-s − i·3-s + 1.90·4-s + 1.47i·5-s − 0.315·6-s − 1.22i·8-s − 9-s + 0.463·10-s + (−1.23 − 3.07i)11-s − 1.90i·12-s + 6.26·13-s + 1.47·15-s + 3.41·16-s − 5.48·17-s + 0.315i·18-s + 1.26·19-s + ⋯ |
L(s) = 1 | − 0.222i·2-s − 0.577i·3-s + 0.950·4-s + 0.658i·5-s − 0.128·6-s − 0.434i·8-s − 0.333·9-s + 0.146·10-s + (−0.372 − 0.927i)11-s − 0.548i·12-s + 1.73·13-s + 0.380·15-s + 0.853·16-s − 1.33·17-s + 0.0742i·18-s + 0.289·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.254110187\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.254110187\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.23 + 3.07i)T \) |
good | 2 | \( 1 + 0.315iT - 2T^{2} \) |
| 5 | \( 1 - 1.47iT - 5T^{2} \) |
| 13 | \( 1 - 6.26T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 + 7.46iT - 29T^{2} \) |
| 31 | \( 1 - 5.39iT - 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 + 7.49iT - 43T^{2} \) |
| 47 | \( 1 - 2.91iT - 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 2.70iT - 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 1.39iT - 79T^{2} \) |
| 83 | \( 1 - 0.851T + 83T^{2} \) |
| 89 | \( 1 + 5.85iT - 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−9.097944031517451656935233911249, −8.420569280089322490504093796597, −7.59511105182838023239108767541, −6.64069529418943289469887914212, −6.35028439379185195896035276184, −5.44754662992577924686004610293, −3.86125739303634724522990056369, −3.05163634991190623942960974274, −2.20258201334855330077331207932, −0.961445415991374787268828088823,
1.32626619380529138717263666064, 2.49437655005356603827364725348, 3.62780877869991983370963077712, 4.61672954507407997713017192445, 5.42192784702306788236777263712, 6.32884428445357120822562597850, 7.03133089340891674634625565713, 8.010983018847682822614521776309, 8.773095136518380351796320964815, 9.358913439281307026549200385520