| L(s) = 1 | + 3-s − 4-s + 9-s − 12-s − 2·13-s + 16-s − 19-s + 6·25-s + 27-s − 2·31-s − 36-s − 6·37-s − 2·39-s − 8·43-s + 48-s − 10·49-s + 2·52-s − 57-s − 12·61-s − 64-s + 16·67-s − 16·73-s + 6·75-s + 76-s − 2·79-s + 81-s − 2·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s + 1/3·9-s − 0.288·12-s − 0.554·13-s + 1/4·16-s − 0.229·19-s + 6/5·25-s + 0.192·27-s − 0.359·31-s − 1/6·36-s − 0.986·37-s − 0.320·39-s − 1.21·43-s + 0.144·48-s − 1.42·49-s + 0.277·52-s − 0.132·57-s − 1.53·61-s − 1/8·64-s + 1.95·67-s − 1.87·73-s + 0.692·75-s + 0.114·76-s − 0.225·79-s + 1/9·81-s − 0.207·93-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
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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
−8.174909159133632079783347581751, −7.61612781027066728090926230059, −7.28710052924718691242519165730, −6.75097531115428009035409284979, −6.37434502731986133989426919565, −5.78488039152357357384257362110, −5.10197391270877893814801944193, −4.83368177898834122124641796928, −4.42487811493704312503751008236, −3.58666975645778525111045986413, −3.35096162577863829375368889427, −2.67137453803310250185603583335, −1.97378436894934694378708018087, −1.25326953558191042500996567214, 0,
1.25326953558191042500996567214, 1.97378436894934694378708018087, 2.67137453803310250185603583335, 3.35096162577863829375368889427, 3.58666975645778525111045986413, 4.42487811493704312503751008236, 4.83368177898834122124641796928, 5.10197391270877893814801944193, 5.78488039152357357384257362110, 6.37434502731986133989426919565, 6.75097531115428009035409284979, 7.28710052924718691242519165730, 7.61612781027066728090926230059, 8.174909159133632079783347581751
