L(s) = 1 | + 3-s − 7-s + 9-s + 4·11-s − 3·13-s + 4·17-s + 19-s − 21-s + 27-s + 8·29-s + 31-s + 4·33-s + 2·37-s − 3·39-s + 2·41-s − 11·43-s + 2·47-s − 6·49-s + 4·51-s − 10·53-s + 57-s + 6·59-s − 11·61-s − 63-s + 9·67-s + 6·71-s + 14·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.832·13-s + 0.970·17-s + 0.229·19-s − 0.218·21-s + 0.192·27-s + 1.48·29-s + 0.179·31-s + 0.696·33-s + 0.328·37-s − 0.480·39-s + 0.312·41-s − 1.67·43-s + 0.291·47-s − 6/7·49-s + 0.560·51-s − 1.37·53-s + 0.132·57-s + 0.781·59-s − 1.40·61-s − 0.125·63-s + 1.09·67-s + 0.712·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.564513158\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.564513158\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−8.198221417997014836494558678482, −7.70330461321421902076177578340, −6.70312208082658114836742787201, −6.40615174728864147495417904450, −5.22932247186022523954200563106, −4.54375322082085705937700478594, −3.57768623982026388309620183029, −3.02041608294368622791977078060, −1.94899645520986314060407705286, −0.886794252557696980979213887488,
0.886794252557696980979213887488, 1.94899645520986314060407705286, 3.02041608294368622791977078060, 3.57768623982026388309620183029, 4.54375322082085705937700478594, 5.22932247186022523954200563106, 6.40615174728864147495417904450, 6.70312208082658114836742787201, 7.70330461321421902076177578340, 8.198221417997014836494558678482