| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·13-s − 17-s − 4·19-s + 4·21-s − 5·25-s − 27-s + 4·31-s − 7·37-s − 4·39-s + 3·41-s − 10·43-s − 6·47-s + 9·49-s + 51-s + 3·53-s + 4·57-s + 9·59-s + 7·61-s − 4·63-s − 2·67-s − 15·71-s + 4·73-s + 5·75-s + 14·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.10·13-s − 0.242·17-s − 0.917·19-s + 0.872·21-s − 25-s − 0.192·27-s + 0.718·31-s − 1.15·37-s − 0.640·39-s + 0.468·41-s − 1.52·43-s − 0.875·47-s + 9/7·49-s + 0.140·51-s + 0.412·53-s + 0.529·57-s + 1.17·59-s + 0.896·61-s − 0.503·63-s − 0.244·67-s − 1.78·71-s + 0.468·73-s + 0.577·75-s + 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.67550575590589, −13.46739533650335, −13.12833593384720, −12.50022328032991, −12.17158154954973, −11.45757000725917, −11.21423592355419, −10.40586430744967, −10.13244077689075, −9.755795327584024, −8.980024832641811, −8.662285309379910, −8.065363225993368, −7.360765038067722, −6.657242726498742, −6.464773135903384, −6.031943382664249, −5.413140777887304, −4.792776637430715, −3.980802889931878, −3.677338487565421, −3.078655549159184, −2.270265125591211, −1.601971396651248, −0.6711820339154943, 0,
0.6711820339154943, 1.601971396651248, 2.270265125591211, 3.078655549159184, 3.677338487565421, 3.980802889931878, 4.792776637430715, 5.413140777887304, 6.031943382664249, 6.464773135903384, 6.657242726498742, 7.360765038067722, 8.065363225993368, 8.662285309379910, 8.980024832641811, 9.755795327584024, 10.13244077689075, 10.40586430744967, 11.21423592355419, 11.45757000725917, 12.17158154954973, 12.50022328032991, 13.12833593384720, 13.46739533650335, 13.67550575590589
