L(s) = 1 | + (−0.646 + 0.762i)2-s + (−0.163 − 0.986i)4-s + (0.992 + 0.119i)5-s + (0.858 + 0.512i)8-s + (−0.733 + 0.680i)10-s + (0.983 + 0.178i)13-s + (−0.946 + 0.323i)16-s + (−0.251 − 0.967i)17-s + (0.669 + 0.743i)19-s + (−0.0448 − 0.998i)20-s + (−0.365 − 0.930i)23-s + (0.971 + 0.237i)25-s + (−0.772 + 0.635i)26-s + (−0.550 + 0.834i)29-s + (0.104 − 0.994i)31-s + (0.365 − 0.930i)32-s + ⋯ |
L(s) = 1 | + (−0.646 + 0.762i)2-s + (−0.163 − 0.986i)4-s + (0.992 + 0.119i)5-s + (0.858 + 0.512i)8-s + (−0.733 + 0.680i)10-s + (0.983 + 0.178i)13-s + (−0.946 + 0.323i)16-s + (−0.251 − 0.967i)17-s + (0.669 + 0.743i)19-s + (−0.0448 − 0.998i)20-s + (−0.365 − 0.930i)23-s + (0.971 + 0.237i)25-s + (−0.772 + 0.635i)26-s + (−0.550 + 0.834i)29-s + (0.104 − 0.994i)31-s + (0.365 − 0.930i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7635844734 - 0.6938953243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7635844734 - 0.6938953243i\) |
\(L(1)\) |
\(\approx\) |
\(0.8591117663 + 0.1501654727i\) |
\(L(1)\) |
\(\approx\) |
\(0.8591117663 + 0.1501654727i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.646 + 0.762i)T \) |
| 5 | \( 1 + (0.992 + 0.119i)T \) |
| 13 | \( 1 + (0.983 + 0.178i)T \) |
| 17 | \( 1 + (-0.251 - 0.967i)T \) |
| 19 | \( 1 + (0.669 + 0.743i)T \) |
| 23 | \( 1 + (-0.365 - 0.930i)T \) |
| 29 | \( 1 + (-0.550 + 0.834i)T \) |
| 31 | \( 1 + (0.104 - 0.994i)T \) |
| 37 | \( 1 + (-0.447 - 0.894i)T \) |
| 41 | \( 1 + (-0.858 - 0.512i)T \) |
| 43 | \( 1 + (0.222 + 0.974i)T \) |
| 47 | \( 1 + (-0.925 - 0.379i)T \) |
| 53 | \( 1 + (-0.712 - 0.701i)T \) |
| 59 | \( 1 + (-0.873 - 0.486i)T \) |
| 61 | \( 1 + (0.712 - 0.701i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.0448 - 0.998i)T \) |
| 73 | \( 1 + (-0.925 + 0.379i)T \) |
| 79 | \( 1 + (-0.913 + 0.406i)T \) |
| 83 | \( 1 + (-0.983 + 0.178i)T \) |
| 89 | \( 1 + (0.826 - 0.563i)T \) |
| 97 | \( 1 + (0.809 + 0.587i)T \) |
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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
−20.463502850767287637334526253691, −19.7466192486111582446973883510, −18.89886193020846104234338094926, −18.19624867494292242958341950077, −17.50990004189107925887635702692, −17.11431372850154585511055465011, −16.09274486734223045479441476385, −15.412618616816643231231891492830, −14.155987516080171699702919504553, −13.34811932972442246435788484646, −13.084015882524970132623411320700, −11.96234733095663652517539535996, −11.26291891131211638776164648158, −10.40699989536896102106369318120, −9.8875772846876963290524623666, −8.96971535676508931829848658222, −8.4837324600697467265062534412, −7.45899995947905525181850897956, −6.4981009932817913960030009157, −5.64828730199434876629109581605, −4.61019640281722823315135802652, −3.56678149965004817439162047968, −2.79390496794530897592393154199, −1.67522470206099464771770170052, −1.20381418670256133768636288193,
0.23189888411448065541484019168, 1.37770232600231661738736182034, 2.10942264217051574260629562632, 3.36578289489118683060897351514, 4.655482784492139149069677896685, 5.46914863339020115965947198923, 6.18164246361652047385063468696, 6.826231011336424712240662595, 7.74664892086825679234610924602, 8.641223908201447030616458195854, 9.33252417424853781421242679777, 9.96243172154195296302983339371, 10.766888569172393630970542737280, 11.49209217460304519516491747364, 12.7708433945057070315184711626, 13.60298288158438810363907057075, 14.17999513967254189079197595165, 14.784028726004760968236808414577, 15.91985570208595962819577649701, 16.32681825176295258952891681423, 17.10801697963121779258081093644, 17.97469678628698182980485722278, 18.39525859123448662878528636805, 18.944394343762097549619712157137, 20.25152403134340102140220770975