| L(s) = 1 | − 8·4-s − 7·7-s − 42·11-s − 20·13-s + 64·16-s + 66·17-s + 38·19-s + 12·23-s + 56·28-s + 258·29-s + 146·31-s − 434·37-s + 282·41-s − 20·43-s + 336·44-s − 72·47-s + 49·49-s + 160·52-s + 336·53-s + 360·59-s − 682·61-s − 512·64-s − 812·67-s − 528·68-s − 810·71-s + 124·73-s − 304·76-s + ⋯ |
| L(s) = 1 | − 4-s − 0.377·7-s − 1.15·11-s − 0.426·13-s + 16-s + 0.941·17-s + 0.458·19-s + 0.108·23-s + 0.377·28-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 + 1/7·49-s + 0.426·52-s + 0.870·53-s + 0.794·59-s − 1.43·61-s − 64-s − 1.48·67-s − 0.941·68-s − 1.35·71-s + 0.198·73-s − 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 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
−8.647693806993091644181500281095, −7.956374849547815156574527860558, −7.22404693537899363648144342117, −6.05423264670412561122702376384, −5.22512260349545614432879817056, −4.62788451128366038680323858364, −3.46250590164234238678405106012, −2.68071685293531656813128191666, −1.06696857213572461580852451823, 0,
1.06696857213572461580852451823, 2.68071685293531656813128191666, 3.46250590164234238678405106012, 4.62788451128366038680323858364, 5.22512260349545614432879817056, 6.05423264670412561122702376384, 7.22404693537899363648144342117, 7.956374849547815156574527860558, 8.647693806993091644181500281095
