| L(s) = 1 | − 8·4-s + 5·5-s − 42·11-s − 20·13-s + 64·16-s + 66·17-s − 38·19-s − 40·20-s − 12·23-s + 25·25-s + 258·29-s − 146·31-s + 434·37-s − 282·41-s + 20·43-s + 336·44-s − 72·47-s + 160·52-s − 336·53-s − 210·55-s − 360·59-s + 682·61-s − 512·64-s − 100·65-s + 812·67-s − 528·68-s − 810·71-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.15·11-s − 0.426·13-s + 16-s + 0.941·17-s − 0.458·19-s − 0.447·20-s − 0.108·23-s + 1/5·25-s + 1.65·29-s − 0.845·31-s + 1.92·37-s − 1.07·41-s + 0.0709·43-s + 1.15·44-s − 0.223·47-s + 0.426·52-s − 0.870·53-s − 0.514·55-s − 0.794·59-s + 1.43·61-s − 64-s − 0.190·65-s + 1.48·67-s − 0.941·68-s − 1.35·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 + 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 - 258 T + p^{3} T^{2} \) |
| 31 | \( 1 + 146 T + p^{3} T^{2} \) |
| 37 | \( 1 - 434 T + p^{3} T^{2} \) |
| 41 | \( 1 + 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 336 T + p^{3} T^{2} \) |
| 59 | \( 1 + 360 T + p^{3} T^{2} \) |
| 61 | \( 1 - 682 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 + 810 T + p^{3} T^{2} \) |
| 73 | \( 1 - 124 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1208 T + p^{3} T^{2} \) |
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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
−8.180845144336725208950192569272, −7.86763727884578752465186768545, −6.71055478766858683568552920170, −5.74030836243527543144812071372, −5.11714673948007374960012805625, −4.42016130412106247171583368918, −3.31092781473497419716077860863, −2.41910258333400064864603127800, −1.08144668099918349951983835532, 0,
1.08144668099918349951983835532, 2.41910258333400064864603127800, 3.31092781473497419716077860863, 4.42016130412106247171583368918, 5.11714673948007374960012805625, 5.74030836243527543144812071372, 6.71055478766858683568552920170, 7.86763727884578752465186768545, 8.180845144336725208950192569272
