| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 6·7-s + 8-s + 9-s + 12-s − 6·14-s + 16-s + 18-s − 6·19-s − 6·21-s + 24-s − 8·25-s + 27-s − 6·28-s + 2·29-s + 32-s + 36-s − 6·38-s − 8·41-s − 6·42-s + 48-s + 14·49-s − 8·50-s − 16·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 2.26·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.60·14-s + 1/4·16-s + 0.235·18-s − 1.37·19-s − 1.30·21-s + 0.204·24-s − 8/5·25-s + 0.192·27-s − 1.13·28-s + 0.371·29-s + 0.176·32-s + 1/6·36-s − 0.973·38-s − 1.24·41-s − 0.925·42-s + 0.144·48-s + 2·49-s − 1.13·50-s − 2.19·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155952 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.171970997359876236438052265917, −8.428894229902090400144328454176, −8.158050168245929904522687302567, −7.43639749873438999689426397380, −6.84735434860807313055611936231, −6.56195078108028705792077305227, −6.07961045769152073480927595381, −5.70878558320159113298410267568, −4.76501231255688371068228955672, −4.25364662443003152722071856521, −3.49312588335554462137764370154, −3.33753716285025543958085057893, −2.56155676231552508176040171181, −1.83013379079956618939329715335, 0,
1.83013379079956618939329715335, 2.56155676231552508176040171181, 3.33753716285025543958085057893, 3.49312588335554462137764370154, 4.25364662443003152722071856521, 4.76501231255688371068228955672, 5.70878558320159113298410267568, 6.07961045769152073480927595381, 6.56195078108028705792077305227, 6.84735434860807313055611936231, 7.43639749873438999689426397380, 8.158050168245929904522687302567, 8.428894229902090400144328454176, 9.171970997359876236438052265917
