L(s) = 1 | + 3-s + 2·5-s + 2·7-s + 9-s + 2·13-s + 2·15-s − 4·17-s + 6·19-s + 2·21-s − 25-s + 27-s + 8·29-s − 8·31-s + 4·35-s + 10·37-s + 2·39-s − 8·41-s + 2·43-s + 2·45-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s + 6·57-s + 12·59-s − 10·61-s + 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.970·17-s + 1.37·19-s + 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.676·35-s + 1.64·37-s + 0.320·39-s − 1.24·41-s + 0.304·43-s + 0.298·45-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s + 0.794·57-s + 1.56·59-s − 1.28·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.750000211\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.750000211\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.555194722852196019830902610758, −8.694963925814389373226655045758, −8.069373725115527033692000658280, −7.13458477543704272109197550389, −6.25340607782487029859951366723, −5.33648935965715263647979723486, −4.50322513346415393072595559425, −3.35247680087092108642086649384, −2.26769467555400519962633969149, −1.33575734148260048331554449892,
1.33575734148260048331554449892, 2.26769467555400519962633969149, 3.35247680087092108642086649384, 4.50322513346415393072595559425, 5.33648935965715263647979723486, 6.25340607782487029859951366723, 7.13458477543704272109197550389, 8.069373725115527033692000658280, 8.694963925814389373226655045758, 9.555194722852196019830902610758