| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 2·11-s − 4·13-s − 14-s + 16-s − 2·17-s − 2·19-s − 2·22-s + 2·23-s + 4·26-s + 28-s + 29-s + 3·31-s − 32-s + 2·34-s + 3·37-s + 2·38-s + 3·41-s − 8·43-s + 2·44-s − 2·46-s + 3·47-s + 49-s − 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.603·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.458·19-s − 0.426·22-s + 0.417·23-s + 0.784·26-s + 0.188·28-s + 0.185·29-s + 0.538·31-s − 0.176·32-s + 0.342·34-s + 0.493·37-s + 0.324·38-s + 0.468·41-s − 1.21·43-s + 0.301·44-s − 0.294·46-s + 0.437·47-s + 1/7·49-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286458876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286458876\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.75017891691249744868306047346, −7.07757229611589348266169688137, −6.53258597990209342786250477418, −5.78463713087148239959559057985, −4.84666270435126237889290760288, −4.33619661515258104848860430615, −3.26637981102033841992381677444, −2.43285556311256985946737489302, −1.68685337663698284936161165780, −0.60447130549484367943743781777,
0.60447130549484367943743781777, 1.68685337663698284936161165780, 2.43285556311256985946737489302, 3.26637981102033841992381677444, 4.33619661515258104848860430615, 4.84666270435126237889290760288, 5.78463713087148239959559057985, 6.53258597990209342786250477418, 7.07757229611589348266169688137, 7.75017891691249744868306047346
