| L(s) = 1 | − 1.03·5-s − 3.86·7-s − 1.46·11-s − 13-s − 5.65·17-s + 1.79·19-s + 6.92·23-s − 3.92·25-s − 2.07·29-s − 3.86·31-s + 3.99·35-s − 2·37-s + 1.03·41-s + 5.65·43-s + 2.53·47-s + 7.92·49-s − 5.65·53-s + 1.51·55-s + 9.46·59-s − 8.92·61-s + 1.03·65-s − 1.79·67-s + 2.53·71-s + 11.8·73-s + 5.65·77-s + 6.53·83-s + 5.85·85-s + ⋯ |
| L(s) = 1 | − 0.462·5-s − 1.46·7-s − 0.441·11-s − 0.277·13-s − 1.37·17-s + 0.411·19-s + 1.44·23-s − 0.785·25-s − 0.384·29-s − 0.693·31-s + 0.676·35-s − 0.328·37-s + 0.161·41-s + 0.862·43-s + 0.369·47-s + 1.13·49-s − 0.777·53-s + 0.204·55-s + 1.23·59-s − 1.14·61-s + 0.128·65-s − 0.219·67-s + 0.300·71-s + 1.38·73-s + 0.644·77-s + 0.717·83-s + 0.635·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8288350074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8288350074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 2.07T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6.53T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.620784353544723525088223480259, −7.61792433877329711683541487527, −7.05633693663306492546509364593, −6.39711759693598498792162993006, −5.56756935449813923606790914244, −4.68047155275216214342668399135, −3.76099057714354351466054334522, −3.06978023499657591366305268137, −2.16826471020983047720537932314, −0.49930122606435192214899492260,
0.49930122606435192214899492260, 2.16826471020983047720537932314, 3.06978023499657591366305268137, 3.76099057714354351466054334522, 4.68047155275216214342668399135, 5.56756935449813923606790914244, 6.39711759693598498792162993006, 7.05633693663306492546509364593, 7.61792433877329711683541487527, 8.620784353544723525088223480259
