L(s) = 1 | + (0.669 + 0.743i)2-s + (0.0627 + 0.998i)3-s + (−0.104 + 0.994i)4-s + (0.387 − 0.921i)5-s + (−0.699 + 0.714i)6-s + (0.996 + 0.0836i)7-s + (−0.809 + 0.587i)8-s + (−0.992 + 0.125i)9-s + (0.944 − 0.328i)10-s + (0.832 + 0.553i)11-s + (−0.999 − 0.0418i)12-s + (−0.348 + 0.937i)13-s + (0.604 + 0.796i)14-s + (0.944 + 0.328i)15-s + (−0.978 − 0.207i)16-s + (−0.570 − 0.821i)17-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)2-s + (0.0627 + 0.998i)3-s + (−0.104 + 0.994i)4-s + (0.387 − 0.921i)5-s + (−0.699 + 0.714i)6-s + (0.996 + 0.0836i)7-s + (−0.809 + 0.587i)8-s + (−0.992 + 0.125i)9-s + (0.944 − 0.328i)10-s + (0.832 + 0.553i)11-s + (−0.999 − 0.0418i)12-s + (−0.348 + 0.937i)13-s + (0.604 + 0.796i)14-s + (0.944 + 0.328i)15-s + (−0.978 − 0.207i)16-s + (−0.570 − 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9591988354 + 1.383485654i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9591988354 + 1.383485654i\) |
\(L(1)\) |
\(\approx\) |
\(1.202486104 + 0.9729723884i\) |
\(L(1)\) |
\(\approx\) |
\(1.202486104 + 0.9729723884i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 151 | \( 1 \) |
good | 2 | \( 1 + (0.669 + 0.743i)T \) |
| 3 | \( 1 + (0.0627 + 0.998i)T \) |
| 5 | \( 1 + (0.387 - 0.921i)T \) |
| 7 | \( 1 + (0.996 + 0.0836i)T \) |
| 11 | \( 1 + (0.832 + 0.553i)T \) |
| 13 | \( 1 + (-0.348 + 0.937i)T \) |
| 17 | \( 1 + (-0.570 - 0.821i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.913 - 0.406i)T \) |
| 29 | \( 1 + (-0.929 - 0.368i)T \) |
| 31 | \( 1 + (-0.957 + 0.289i)T \) |
| 37 | \( 1 + (0.985 - 0.166i)T \) |
| 41 | \( 1 + (0.968 + 0.248i)T \) |
| 43 | \( 1 + (0.996 - 0.0836i)T \) |
| 47 | \( 1 + (-0.268 - 0.963i)T \) |
| 53 | \( 1 + (0.535 + 0.844i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.895 + 0.444i)T \) |
| 67 | \( 1 + (-0.992 - 0.125i)T \) |
| 71 | \( 1 + (-0.570 + 0.821i)T \) |
| 73 | \( 1 + (-0.425 - 0.904i)T \) |
| 79 | \( 1 + (0.728 - 0.684i)T \) |
| 83 | \( 1 + (0.876 - 0.481i)T \) |
| 89 | \( 1 + (-0.855 - 0.518i)T \) |
| 97 | \( 1 + (0.604 - 0.796i)T \) |
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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
−27.86658750233366537362156687980, −27.04206890877585074689111711541, −25.51822216933146787090551556052, −24.59148355920164656579660161924, −23.80517732555394721765498839039, −22.75286489347103215604208128236, −21.963185727440706920591839139486, −20.896837125548646794265551086522, −19.73108123194625486935275788174, −18.97843653021864962146549241203, −18.00557046692380115680131082844, −17.199623852463465945415912303952, −14.7748308529216571100505962847, −14.66184429373034741764899494161, −13.43620505726719279294286846827, −12.56721455628567648677129713102, −11.22500323309256682752476953749, −10.82652876961973252233879351349, −9.137127621241662530424959393, −7.70110149348817380726750855267, −6.39517886502305057167380772817, −5.536656525109091126462052435864, −3.76807304417996391179730374295, −2.46742545188370684766465068608, −1.42858331402699211497507398540,
2.28831594144626107283289563596, 4.34205130262080966573660005527, 4.60360016766279797104936105368, 5.818734856972443219137628947099, 7.31386594924064814835204793902, 8.89241570535633390382895213207, 9.1819562625038843350088387988, 11.14839292472806268218637415976, 12.05863126012647910525114760825, 13.40196713368923840975335527971, 14.48739255906880121939248265738, 15.09040743660898422676659708190, 16.40939206908503711067065479265, 16.98555843229332489650185762213, 17.84765841579490890170719856026, 19.920809642211444530209959098815, 20.859920362245316917504150792621, 21.47277161384581029162218220192, 22.36088673702397194061965771387, 23.54517098790999711954042320078, 24.54442284118215786741909402550, 25.22406843884982556558443895360, 26.355427538602326105833056622463, 27.31155889789618409261855434340, 28.080307791951937215582281187958