L(s) = 1 | + (0.913 + 0.406i)2-s + (0.728 + 0.684i)3-s + (0.669 + 0.743i)4-s + (0.832 − 0.553i)5-s + (0.387 + 0.921i)6-s + (−0.999 − 0.0418i)7-s + (0.309 + 0.951i)8-s + (0.0627 + 0.998i)9-s + (0.985 − 0.166i)10-s + (−0.957 − 0.289i)11-s + (−0.0209 + 0.999i)12-s + (−0.570 − 0.821i)13-s + (−0.895 − 0.444i)14-s + (0.985 + 0.166i)15-s + (−0.104 + 0.994i)16-s + (0.463 − 0.886i)17-s + ⋯ |
L(s) = 1 | + (0.913 + 0.406i)2-s + (0.728 + 0.684i)3-s + (0.669 + 0.743i)4-s + (0.832 − 0.553i)5-s + (0.387 + 0.921i)6-s + (−0.999 − 0.0418i)7-s + (0.309 + 0.951i)8-s + (0.0627 + 0.998i)9-s + (0.985 − 0.166i)10-s + (−0.957 − 0.289i)11-s + (−0.0209 + 0.999i)12-s + (−0.570 − 0.821i)13-s + (−0.895 − 0.444i)14-s + (0.985 + 0.166i)15-s + (−0.104 + 0.994i)16-s + (0.463 − 0.886i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.478 + 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.928775880 + 1.144978738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.928775880 + 1.144978738i\) |
\(L(1)\) |
\(\approx\) |
\(1.853891190 + 0.7630911120i\) |
\(L(1)\) |
\(\approx\) |
\(1.853891190 + 0.7630911120i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 151 | \( 1 \) |
good | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 3 | \( 1 + (0.728 + 0.684i)T \) |
| 5 | \( 1 + (0.832 - 0.553i)T \) |
| 7 | \( 1 + (-0.999 - 0.0418i)T \) |
| 11 | \( 1 + (-0.957 - 0.289i)T \) |
| 13 | \( 1 + (-0.570 - 0.821i)T \) |
| 17 | \( 1 + (0.463 - 0.886i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.978 + 0.207i)T \) |
| 29 | \( 1 + (-0.187 + 0.982i)T \) |
| 31 | \( 1 + (0.146 + 0.989i)T \) |
| 37 | \( 1 + (0.996 - 0.0836i)T \) |
| 41 | \( 1 + (-0.992 - 0.125i)T \) |
| 43 | \( 1 + (-0.999 + 0.0418i)T \) |
| 47 | \( 1 + (0.604 - 0.796i)T \) |
| 53 | \( 1 + (0.876 + 0.481i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (0.228 + 0.973i)T \) |
| 67 | \( 1 + (0.0627 - 0.998i)T \) |
| 71 | \( 1 + (0.463 + 0.886i)T \) |
| 73 | \( 1 + (0.535 - 0.844i)T \) |
| 79 | \( 1 + (-0.929 + 0.368i)T \) |
| 83 | \( 1 + (0.968 - 0.248i)T \) |
| 89 | \( 1 + (-0.268 + 0.963i)T \) |
| 97 | \( 1 + (-0.895 + 0.444i)T \) |
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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
−28.56389752329593830360818418041, −26.45482887921772572007787821408, −25.79382850967436255309315719506, −24.94694241843311692591824720837, −23.897825901736616117510658189215, −22.98758644089557818954543053497, −21.93111034628257381528188318803, −21.05114002545688479496879561160, −20.07638538601618060440263932161, −18.95861953676829760642846904852, −18.47844254309329243903360132929, −16.79173212567832260875033161988, −15.36045978496273613624075431385, −14.473317426201417485169580555615, −13.57319687937695538811883439471, −12.86689067976545666247078491391, −11.89527397104021886248530395254, −10.15802403508718373407529090224, −9.66588109675863615939882921047, −7.728751291246156889769323527703, −6.54027314649018135025939008260, −5.7933927578014927058481418014, −3.890771227620854631070112523320, −2.71013400794593067471483673643, −1.8902725744889215843966845781,
2.50955303571923716358118448129, 3.286072370743719282744758564776, 4.902150287134351659722704456736, 5.57918504544601546520192112261, 7.150351931869424812657148598472, 8.38960971941435100006342182193, 9.61929783340504944046994640726, 10.54912820327268652591012295770, 12.33076159735282236127474271383, 13.34732749893590500703934028095, 13.84449741308753619737643091323, 15.177932962344251719012130598994, 16.05086367526025817530952956338, 16.69093378081316359114535625851, 18.13553680160163456578921359801, 19.93699251473531263043181369936, 20.36709348827211278776477492092, 21.64366009955496582601950046411, 22.0051180986343573044876604254, 23.271229828855238840177215587412, 24.508853658590303605707251363543, 25.36069338744487975941335507102, 25.91838766248266519175936355914, 26.90137478608344653167183909661, 28.42148605293766082640819168940