| L(s) = 1 | + 2·2-s − 3-s + 3·4-s − 2·6-s + 4·8-s + 9-s − 3·12-s + 5·16-s + 2·18-s − 19-s − 4·24-s − 6·25-s − 27-s − 4·29-s + 6·32-s + 3·36-s − 2·38-s + 20·41-s + 8·43-s − 5·48-s − 14·49-s − 12·50-s − 20·53-s − 2·54-s + 57-s − 8·58-s + 24·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.816·6-s + 1.41·8-s + 1/3·9-s − 0.866·12-s + 5/4·16-s + 0.471·18-s − 0.229·19-s − 0.816·24-s − 6/5·25-s − 0.192·27-s − 0.742·29-s + 1.06·32-s + 1/2·36-s − 0.324·38-s + 3.12·41-s + 1.21·43-s − 0.721·48-s − 2·49-s − 1.69·50-s − 2.74·53-s − 0.272·54-s + 0.132·57-s − 1.05·58-s + 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740772 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.081206648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.081206648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.987570621007900466899230715503, −7.84351324830482614195159853030, −7.19583741265591956235027568701, −6.73529275791267126360213503833, −6.46925034993373182658971924423, −5.70322294388590675424631045525, −5.66755609194805589043695166783, −5.20173444424979461085997861151, −4.39790054298019034493919320980, −4.21125984248492975559254303317, −3.72105155927604818995516413488, −3.06380200487442055779450117560, −2.33973473634575001863443152397, −1.89717432081133819255264343684, −0.816282240239440914761831766450,
0.816282240239440914761831766450, 1.89717432081133819255264343684, 2.33973473634575001863443152397, 3.06380200487442055779450117560, 3.72105155927604818995516413488, 4.21125984248492975559254303317, 4.39790054298019034493919320980, 5.20173444424979461085997861151, 5.66755609194805589043695166783, 5.70322294388590675424631045525, 6.46925034993373182658971924423, 6.73529275791267126360213503833, 7.19583741265591956235027568701, 7.84351324830482614195159853030, 7.987570621007900466899230715503
