| L(s) = 1 | + 1.51·2-s + 3-s + 0.296·4-s + 1.13·5-s + 1.51·6-s + 2.34·7-s − 2.58·8-s + 9-s + 1.71·10-s + 4.67·11-s + 0.296·12-s + 4.73·13-s + 3.55·14-s + 1.13·15-s − 4.50·16-s + 17-s + 1.51·18-s − 1.43·19-s + 0.336·20-s + 2.34·21-s + 7.09·22-s + 1.75·23-s − 2.58·24-s − 3.71·25-s + 7.18·26-s + 27-s + 0.695·28-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.577·3-s + 0.148·4-s + 0.507·5-s + 0.618·6-s + 0.886·7-s − 0.912·8-s + 0.333·9-s + 0.543·10-s + 1.41·11-s + 0.0855·12-s + 1.31·13-s + 0.950·14-s + 0.292·15-s − 1.12·16-s + 0.242·17-s + 0.357·18-s − 0.329·19-s + 0.0752·20-s + 0.511·21-s + 1.51·22-s + 0.366·23-s − 0.526·24-s − 0.742·25-s + 1.40·26-s + 0.192·27-s + 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.174634355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.174634355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.51T + 2T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 - 4.73T + 13T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 3.63T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 - 7.11T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 9.83T + 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 0.981T + 71T^{2} \) |
| 73 | \( 1 + 8.45T + 73T^{2} \) |
| 83 | \( 1 + 0.0563T + 83T^{2} \) |
| 89 | \( 1 + 4.99T + 89T^{2} \) |
| 97 | \( 1 + 5.59T + 97T^{2} \) |
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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
−8.579949577965290638411409337099, −7.72302292124058195451042454768, −6.76785644858871390372846259107, −5.97263301955874523403401054206, −5.55193905062873077632779741353, −4.37288901341984089804649407346, −4.02157034487732861951622440959, −3.23231864376862219627989335177, −2.08473497023800322561886335409, −1.22388629372659223769787181501,
1.22388629372659223769787181501, 2.08473497023800322561886335409, 3.23231864376862219627989335177, 4.02157034487732861951622440959, 4.37288901341984089804649407346, 5.55193905062873077632779741353, 5.97263301955874523403401054206, 6.76785644858871390372846259107, 7.72302292124058195451042454768, 8.579949577965290638411409337099
