L(s) = 1 | + 2.59i·3-s + (−1.73 + 1.41i)5-s + 3.38·7-s − 3.73·9-s − 1.75i·11-s + 6.21·13-s + (−3.66 − 4.49i)15-s + 6.69i·17-s + (1.34 + 4.14i)19-s + 8.78i·21-s − 5.86·23-s + (0.999 − 4.89i)25-s − 1.89i·27-s + 6.43i·29-s + 6.35·31-s + ⋯ |
L(s) = 1 | + 1.49i·3-s + (−0.774 + 0.632i)5-s + 1.27·7-s − 1.24·9-s − 0.528i·11-s + 1.72·13-s + (−0.947 − 1.16i)15-s + 1.62i·17-s + (0.308 + 0.951i)19-s + 1.91i·21-s − 1.22·23-s + (0.199 − 0.979i)25-s − 0.365i·27-s + 1.19i·29-s + 1.14·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.840 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.702784339\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702784339\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.73 - 1.41i)T \) |
| 19 | \( 1 + (-1.34 - 4.14i)T \) |
good | 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 + 1.75iT - 11T^{2} \) |
| 13 | \( 1 - 6.21T + 13T^{2} \) |
| 17 | \( 1 - 6.69iT - 17T^{2} \) |
| 23 | \( 1 + 5.86T + 23T^{2} \) |
| 29 | \( 1 - 6.43iT - 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 - 1.66T + 37T^{2} \) |
| 41 | \( 1 + 8.78iT - 41T^{2} \) |
| 43 | \( 1 - 0.907T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.35T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 4.49iT - 67T^{2} \) |
| 71 | \( 1 - 1.70T + 71T^{2} \) |
| 73 | \( 1 + 3.58iT - 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 1.57T + 83T^{2} \) |
| 89 | \( 1 + 6.43iT - 89T^{2} \) |
| 97 | \( 1 + 1.66T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19233928012062169220435574939, −8.674492722748558178024236489681, −8.467599619570361704043141534784, −7.71310190137936040309491420906, −6.28516870075620434787300033071, −5.64768131751798574449522359431, −4.50772918956734877220388187602, −3.84665658735849446909578109537, −3.35810302511075068844708383718, −1.59336666694756866006143781536,
0.75661014410970127762747606268, 1.54376686299749598365151869130, 2.72475179375020483845198940593, 4.20082281256948226314532437650, 4.92476146455796386402574254827, 6.00807455235315795119799039996, 6.88789852548244290297378676073, 7.71921794120592343185266497461, 8.113268460215946148761725939228, 8.741634510126315667996510263335