L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (0.0990 − 0.433i)6-s + (0.623 + 0.781i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (−0.222 + 0.974i)10-s + (−0.277 + 0.347i)12-s + (−0.222 − 0.974i)14-s + (0.400 − 0.193i)15-s + (−0.222 + 0.974i)16-s − 0.801·18-s + (0.623 − 0.781i)20-s + (−0.277 + 0.347i)21-s + ⋯ |
L(s) = 1 | + (−0.900 − 0.433i)2-s + (0.0990 + 0.433i)3-s + (0.623 + 0.781i)4-s + (−0.222 − 0.974i)5-s + (0.0990 − 0.433i)6-s + (0.623 + 0.781i)7-s + (−0.222 − 0.974i)8-s + (0.722 − 0.347i)9-s + (−0.222 + 0.974i)10-s + (−0.277 + 0.347i)12-s + (−0.222 − 0.974i)14-s + (0.400 − 0.193i)15-s + (−0.222 + 0.974i)16-s − 0.801·18-s + (0.623 − 0.781i)20-s + (−0.277 + 0.347i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7687542335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7687542335\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 7 | \( 1 + (-0.623 - 0.781i)T \) |
good | 3 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 17 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 29 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 41 | \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 59 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 61 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.880719132440720311692941972912, −9.406368714436869079322194954851, −8.521427136806968430495173493501, −8.093174568304509132996504930576, −7.02114661867174166415869748774, −5.85516602829611368250019684451, −4.66832957070530817160458238953, −3.90920164279983503614332235459, −2.49324842367779469656759081312, −1.25378715259979834882786097227,
1.36953653618641335292831249047, 2.53665395278424799020624244309, 4.00798005204198244153932898163, 5.20956499084148179846986641321, 6.45870958666380709779237378548, 7.07573383618858089521821062805, 7.67945246939936311902600582240, 8.290594002429078436627385673287, 9.483889198040157227647660658112, 10.35008412626855364785913052243