L(s) = 1 | + (0.778 + 0.628i)2-s + (−0.828 + 0.559i)3-s + (0.210 + 0.977i)4-s + (0.0424 + 0.999i)5-s + (−0.996 − 0.0848i)6-s + (−0.594 + 0.803i)7-s + (−0.450 + 0.892i)8-s + (0.372 − 0.927i)9-s + (−0.594 + 0.803i)10-s + (0.210 − 0.977i)11-s + (−0.721 − 0.691i)12-s + (−0.127 − 0.991i)13-s + (−0.967 + 0.251i)14-s + (−0.594 − 0.803i)15-s + (−0.911 + 0.411i)16-s + (−0.828 + 0.559i)17-s + ⋯ |
L(s) = 1 | + (0.778 + 0.628i)2-s + (−0.828 + 0.559i)3-s + (0.210 + 0.977i)4-s + (0.0424 + 0.999i)5-s + (−0.996 − 0.0848i)6-s + (−0.594 + 0.803i)7-s + (−0.450 + 0.892i)8-s + (0.372 − 0.927i)9-s + (−0.594 + 0.803i)10-s + (0.210 − 0.977i)11-s + (−0.721 − 0.691i)12-s + (−0.127 − 0.991i)13-s + (−0.967 + 0.251i)14-s + (−0.594 − 0.803i)15-s + (−0.911 + 0.411i)16-s + (−0.828 + 0.559i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1411087572 + 1.085961279i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1411087572 + 1.085961279i\) |
\(L(1)\) |
\(\approx\) |
\(0.7206010692 + 0.8408893251i\) |
\(L(1)\) |
\(\approx\) |
\(0.7206010692 + 0.8408893251i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 149 | \( 1 \) |
good | 2 | \( 1 + (0.778 + 0.628i)T \) |
| 3 | \( 1 + (-0.828 + 0.559i)T \) |
| 5 | \( 1 + (0.0424 + 0.999i)T \) |
| 7 | \( 1 + (-0.594 + 0.803i)T \) |
| 11 | \( 1 + (0.210 - 0.977i)T \) |
| 13 | \( 1 + (-0.127 - 0.991i)T \) |
| 17 | \( 1 + (-0.828 + 0.559i)T \) |
| 19 | \( 1 + (0.873 + 0.487i)T \) |
| 23 | \( 1 + (-0.127 + 0.991i)T \) |
| 29 | \( 1 + (0.985 - 0.169i)T \) |
| 31 | \( 1 + (-0.127 + 0.991i)T \) |
| 37 | \( 1 + (0.210 - 0.977i)T \) |
| 41 | \( 1 + (-0.911 + 0.411i)T \) |
| 43 | \( 1 + (0.942 + 0.333i)T \) |
| 47 | \( 1 + (0.873 - 0.487i)T \) |
| 53 | \( 1 + (0.942 - 0.333i)T \) |
| 59 | \( 1 + (0.524 + 0.851i)T \) |
| 61 | \( 1 + (0.778 + 0.628i)T \) |
| 67 | \( 1 + (-0.967 - 0.251i)T \) |
| 71 | \( 1 + (0.0424 + 0.999i)T \) |
| 73 | \( 1 + (-0.292 + 0.956i)T \) |
| 79 | \( 1 + (-0.292 - 0.956i)T \) |
| 83 | \( 1 + (-0.292 + 0.956i)T \) |
| 89 | \( 1 + (0.660 - 0.750i)T \) |
| 97 | \( 1 + (0.942 - 0.333i)T \) |
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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
−28.2488053775777247821788834047, −27.00823932647431208998171372904, −25.34041330615883981892037553465, −24.239769081907123884913370544772, −23.75073673260420104023878168124, −22.69444348331725776930529514548, −22.03956597087488333662170074838, −20.57977506117210789110522480072, −19.98209232131659491692890370771, −18.908966323642025474242377463957, −17.63298149705347669173763636888, −16.57946281565572355183673883986, −15.70301560610170847851441379833, −13.937804178364327312059297189013, −13.24341341126469214992311469317, −12.30036664837896235736589312895, −11.589666478773121401018250943763, −10.27836076815579092893564138754, −9.266146664472453654235418714615, −7.21533321322006221599445422077, −6.36326954338233046923949852290, −4.88740882759229843030234599804, −4.27544239669899372624163384245, −2.1883451729559549238635134007, −0.8318423284297148312575059396,
2.90182580055141637273334932590, 3.78354619694189997996195186396, 5.51902722810709425207982796158, 6.02253321274133484047692529786, 7.12993182575862185468060109032, 8.69553964925299720647924202486, 10.172213454569298782505844851872, 11.30786148900449525005012318460, 12.17228078959245945330703509189, 13.395661849452561777284739317747, 14.64987264869332577141682490141, 15.57972774201792458061061352374, 16.12224336555552845282658687777, 17.50073701410695691445165730151, 18.20839143493859432703089150011, 19.67864659285036868845761824925, 21.35910141280282479260291786438, 21.93488000038099611442762701747, 22.56499867832650079606261553358, 23.38287509304784566513131179670, 24.601234111231627551588372241779, 25.54976314534637848345955145671, 26.65105050322207624985849223562, 27.25337703792069075387520027308, 28.80920755892593390574963179200