| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 1.65·5-s + 2.24·8-s − 2.97·10-s + 11-s + 1.80·16-s + 1.91·19-s − 3.71·20-s + 1.80·22-s − 0.730·23-s + 1.73·25-s + 1.97·29-s + 1.00·32-s + 3.44·38-s − 3.71·40-s − 1.24·41-s + 2.24·44-s − 1.31·46-s + 49-s + 3.11·50-s − 1.91·53-s − 1.65·55-s + 3.56·58-s − 0.149·59-s − 0.445·61-s + 4.29·76-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s − 1.65·5-s + 2.24·8-s − 2.97·10-s + 11-s + 1.80·16-s + 1.91·19-s − 3.71·20-s + 1.80·22-s − 0.730·23-s + 1.73·25-s + 1.97·29-s + 1.00·32-s + 3.44·38-s − 3.71·40-s − 1.24·41-s + 2.24·44-s − 1.31·46-s + 49-s + 3.11·50-s − 1.91·53-s − 1.65·55-s + 3.56·58-s − 0.149·59-s − 0.445·61-s + 4.29·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.208850183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.208850183\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 431 | \( 1 + T \) |
| good | 2 | \( 1 - 1.80T + T^{2} \) |
| 5 | \( 1 + 1.65T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.91T + T^{2} \) |
| 23 | \( 1 + 0.730T + T^{2} \) |
| 29 | \( 1 - 1.97T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.24T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.91T + T^{2} \) |
| 59 | \( 1 + 0.149T + T^{2} \) |
| 61 | \( 1 + 0.445T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 1.97T + T^{2} \) |
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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
−8.303014650692079790886097706222, −7.69331149357877972002465870862, −6.94648432749113379492023692568, −6.44756897858453679852076715874, −5.43471539915241881118064742335, −4.68346617134386963788478081030, −4.10349878233346116816616237061, −3.41585109624436285027587725635, −2.87185154315713779054875632805, −1.32205620022166608453593640824,
1.32205620022166608453593640824, 2.87185154315713779054875632805, 3.41585109624436285027587725635, 4.10349878233346116816616237061, 4.68346617134386963788478081030, 5.43471539915241881118064742335, 6.44756897858453679852076715874, 6.94648432749113379492023692568, 7.69331149357877972002465870862, 8.303014650692079790886097706222
