L(s) = 1 | + (−1.37 + 0.331i)2-s + (1.78 − 0.910i)4-s + (1.64 + 1.51i)5-s + 0.936·7-s + (−2.14 + 1.84i)8-s + (−2.76 − 1.53i)10-s + 2.20i·11-s + 3.33·13-s + (−1.28 + 0.310i)14-s + (2.34 − 3.24i)16-s − 1.54·17-s − 3.12·19-s + (4.31 + 1.18i)20-s + (−0.731 − 3.03i)22-s − 3.39i·23-s + ⋯ |
L(s) = 1 | + (−0.972 + 0.234i)2-s + (0.890 − 0.455i)4-s + (0.737 + 0.675i)5-s + 0.353·7-s + (−0.759 + 0.650i)8-s + (−0.875 − 0.483i)10-s + 0.665i·11-s + 0.924·13-s + (−0.344 + 0.0828i)14-s + (0.585 − 0.810i)16-s − 0.374·17-s − 0.716·19-s + (0.964 + 0.265i)20-s + (−0.155 − 0.647i)22-s − 0.707i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.930989 + 0.462834i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.930989 + 0.462834i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 - 0.331i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.64 - 1.51i)T \) |
good | 7 | \( 1 - 0.936T + 7T^{2} \) |
| 11 | \( 1 - 2.20iT - 11T^{2} \) |
| 13 | \( 1 - 3.33T + 13T^{2} \) |
| 17 | \( 1 + 1.54T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 + 3.39iT - 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 8.30iT - 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 - 7.77iT - 43T^{2} \) |
| 47 | \( 1 + 10.7iT - 47T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 5.59iT - 73T^{2} \) |
| 79 | \( 1 - 1.02iT - 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 + 13.0iT - 89T^{2} \) |
| 97 | \( 1 + 2.18iT - 97T^{2} \) |
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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
−11.15399833685619417019131939118, −10.56655776140896249823488319813, −9.785213533647176176815376471047, −8.791794066790340558251374376177, −7.965933928964309246935863025603, −6.68535945384832595423293101179, −6.27417417565894895621362699591, −4.81288647588673909455643605621, −2.88542075493203085944204256910, −1.62646792372811048951692543950,
1.10666746908855797198803650685, 2.49001092251217997144707563880, 4.12552380598186158490468204218, 5.72379553783218024242215926163, 6.49581414082637806121756328410, 7.925360457674972738311337989531, 8.632195492127046384920777036026, 9.344844670648950130854113286643, 10.35224192307352974266320962456, 11.13517502896787436159661703667