| L(s) = 1 | − 2·4-s − 3·5-s + 2·7-s + 3·11-s + 2·13-s + 4·16-s − 17-s + 5·19-s + 6·20-s + 4·25-s − 4·28-s + 3·29-s + 8·31-s − 6·35-s + 8·37-s − 6·41-s − 4·43-s − 6·44-s + 6·47-s − 3·49-s − 4·52-s − 12·53-s − 9·55-s + 12·59-s − 10·61-s − 8·64-s − 6·65-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s + 0.755·7-s + 0.904·11-s + 0.554·13-s + 16-s − 0.242·17-s + 1.14·19-s + 1.34·20-s + 4/5·25-s − 0.755·28-s + 0.557·29-s + 1.43·31-s − 1.01·35-s + 1.31·37-s − 0.937·41-s − 0.609·43-s − 0.904·44-s + 0.875·47-s − 3/7·49-s − 0.554·52-s − 1.64·53-s − 1.21·55-s + 1.56·59-s − 1.28·61-s − 64-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014828892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014828892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−11.38093182139037659842566820047, −10.10278302014150851075341421887, −9.109300011117320213307134485881, −8.256713755120548244557004000097, −7.74542291215750154959689622231, −6.44257768768837718607648442666, −5.01146682876845260799537247768, −4.25748270915351505319381939602, −3.37512184396728216346992291538, −1.01021305890200635033573864009,
1.01021305890200635033573864009, 3.37512184396728216346992291538, 4.25748270915351505319381939602, 5.01146682876845260799537247768, 6.44257768768837718607648442666, 7.74542291215750154959689622231, 8.256713755120548244557004000097, 9.109300011117320213307134485881, 10.10278302014150851075341421887, 11.38093182139037659842566820047
