L(s) = 1 | + (0.984 + 0.175i)2-s + (0.938 + 0.346i)4-s + (−0.612 − 0.790i)5-s + (0.862 + 0.505i)8-s + (−0.464 − 0.885i)10-s + (0.963 + 0.268i)11-s + (−0.591 − 0.806i)13-s + (0.760 + 0.649i)16-s + (−0.169 + 0.985i)17-s + (−0.580 − 0.814i)19-s + (−0.301 − 0.953i)20-s + (0.900 + 0.433i)22-s + (−0.248 + 0.968i)25-s + (−0.440 − 0.897i)26-s + (0.768 + 0.639i)29-s + ⋯ |
L(s) = 1 | + (0.984 + 0.175i)2-s + (0.938 + 0.346i)4-s + (−0.612 − 0.790i)5-s + (0.862 + 0.505i)8-s + (−0.464 − 0.885i)10-s + (0.963 + 0.268i)11-s + (−0.591 − 0.806i)13-s + (0.760 + 0.649i)16-s + (−0.169 + 0.985i)17-s + (−0.580 − 0.814i)19-s + (−0.301 − 0.953i)20-s + (0.900 + 0.433i)22-s + (−0.248 + 0.968i)25-s + (−0.440 − 0.897i)26-s + (0.768 + 0.639i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.199004950 - 0.3131929917i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.199004950 - 0.3131929917i\) |
\(L(1)\) |
\(\approx\) |
\(1.874260784 + 0.02082466407i\) |
\(L(1)\) |
\(\approx\) |
\(1.874260784 + 0.02082466407i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.984 + 0.175i)T \) |
| 5 | \( 1 + (-0.612 - 0.790i)T \) |
| 11 | \( 1 + (0.963 + 0.268i)T \) |
| 13 | \( 1 + (-0.591 - 0.806i)T \) |
| 17 | \( 1 + (-0.169 + 0.985i)T \) |
| 19 | \( 1 + (-0.580 - 0.814i)T \) |
| 29 | \( 1 + (0.768 + 0.639i)T \) |
| 31 | \( 1 + (-0.786 + 0.618i)T \) |
| 37 | \( 1 + (0.352 - 0.935i)T \) |
| 41 | \( 1 + (0.377 - 0.925i)T \) |
| 43 | \( 1 + (0.862 - 0.505i)T \) |
| 47 | \( 1 + (0.733 + 0.680i)T \) |
| 53 | \( 1 + (0.704 - 0.709i)T \) |
| 59 | \( 1 + (-0.464 - 0.885i)T \) |
| 61 | \( 1 + (0.997 - 0.0679i)T \) |
| 67 | \( 1 + (0.888 + 0.458i)T \) |
| 71 | \( 1 + (0.992 - 0.122i)T \) |
| 73 | \( 1 + (0.314 + 0.949i)T \) |
| 79 | \( 1 + (0.327 + 0.945i)T \) |
| 83 | \( 1 + (-0.0611 + 0.998i)T \) |
| 89 | \( 1 + (-0.427 + 0.903i)T \) |
| 97 | \( 1 + (0.959 - 0.281i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.86320886491960247684888518515, −18.53624243551938599215219336682, −17.26592289891809715626805952605, −16.572999368418431564900111039323, −15.99362616118610292156270545305, −15.127937794854341019636663863298, −14.600985116088535301085465055103, −14.10838251095415176829459952212, −13.43954572956362718285553458798, −12.418297653721807655899230216947, −11.750161187065367051844148962595, −11.495792847261225642361310829512, −10.61950635100090409059199725008, −9.87425353481454722215818378426, −9.0531516415340165975214691913, −7.87599278630448978887540049134, −7.29007741529889217230787870396, −6.47989765196645918111515319238, −6.109024126205012398373688131665, −4.90525021274980360894878654928, −4.19268846291082567225139395727, −3.68584900185247604557224062786, −2.714630384697398578373441493210, −2.12703606352901334153546015693, −0.92304978397883727066991104981,
0.815751811449847394446427008671, 1.85701495744166129085319159206, 2.72884573032078807932196978617, 3.90587833927368948759417028857, 4.071698519246426007582163448795, 5.12184295700203180269753260551, 5.59099446290741512176656744822, 6.69557624310828137293776056136, 7.17970590955820101387286285153, 8.11029805965560097356658479759, 8.69899288199132540688856351240, 9.58304491905944297319595046522, 10.751115346445602560030159755758, 11.13361374646510798622079562368, 12.22531910392113998135342506537, 12.526919841171737651075154709237, 12.98915834043248839480794256737, 14.083545532974430194783388237581, 14.60331250982365706271756307310, 15.39605150797981117447889927810, 15.78231067796901682172124414612, 16.68746239086034025349931081756, 17.24082657112083931348917045815, 17.76538066429620653194252467551, 19.301754621084760684048614287395