| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 3·7-s − 8-s + 9-s + 10-s + 6·11-s + 12-s + 13-s + 3·14-s − 15-s + 16-s − 18-s − 5·19-s − 20-s − 3·21-s − 6·22-s − 6·23-s − 24-s + 25-s − 26-s + 27-s − 3·28-s + 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.14·19-s − 0.223·20-s − 0.654·21-s − 1.27·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.566·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.578614365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.578614365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.43360246362285, −12.00076430649624, −11.60599127916108, −11.21704286837200, −10.46983909189112, −10.00828848390986, −9.898587809858144, −9.243322450827480, −8.804440903438034, −8.505203377983676, −8.126904691349768, −7.446488351957575, −6.957160775697346, −6.511774630564765, −6.250485365368798, −5.849576296039178, −4.789964039713790, −4.307524544430275, −3.813392205724137, −3.450562070606184, −2.937277634080123, −2.102130483249474, −1.872099193555508, −0.8234235414442188, −0.5963870470496516,
0.5963870470496516, 0.8234235414442188, 1.872099193555508, 2.102130483249474, 2.937277634080123, 3.450562070606184, 3.813392205724137, 4.307524544430275, 4.789964039713790, 5.849576296039178, 6.250485365368798, 6.511774630564765, 6.957160775697346, 7.446488351957575, 8.126904691349768, 8.505203377983676, 8.804440903438034, 9.243322450827480, 9.898587809858144, 10.00828848390986, 10.46983909189112, 11.21704286837200, 11.60599127916108, 12.00076430649624, 12.43360246362285
