| L(s) = 1 | − 2-s − 3·4-s − 5-s − 7-s + 5·8-s − 9-s + 10-s + 9·11-s + 14-s + 5·16-s + 18-s + 3·20-s − 9·22-s + 2·23-s + 3·25-s + 3·28-s − 15·32-s + 35-s + 3·36-s − 5·40-s − 13·41-s − 27·44-s + 45-s − 2·46-s − 6·49-s − 3·50-s + 8·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 3/2·4-s − 0.447·5-s − 0.377·7-s + 1.76·8-s − 1/3·9-s + 0.316·10-s + 2.71·11-s + 0.267·14-s + 5/4·16-s + 0.235·18-s + 0.670·20-s − 1.91·22-s + 0.417·23-s + 3/5·25-s + 0.566·28-s − 2.65·32-s + 0.169·35-s + 1/2·36-s − 0.790·40-s − 2.03·41-s − 4.07·44-s + 0.149·45-s − 0.294·46-s − 6/7·49-s − 0.424·50-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411207 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411207 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6901060152\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6901060152\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 449 | $C_2$ | \( 1 + 15 T + p T^{2} \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 76 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.990453761305537290238286978106, −7.68760088927311613785086921017, −7.11612916915688730116524632128, −6.72702583523798218996067462243, −6.20996380136793041156757887995, −5.92386563910971412800333863005, −4.97757425313137413806387389994, −4.87503032442712704542067175505, −4.29565723670981607229174458647, −3.83306742079194062572550856840, −3.49587644045599407553253924814, −3.01170620562196387579699425092, −1.66561709229459183966990535634, −1.32348864832389091331280318607, −0.47512840315950026489597492862,
0.47512840315950026489597492862, 1.32348864832389091331280318607, 1.66561709229459183966990535634, 3.01170620562196387579699425092, 3.49587644045599407553253924814, 3.83306742079194062572550856840, 4.29565723670981607229174458647, 4.87503032442712704542067175505, 4.97757425313137413806387389994, 5.92386563910971412800333863005, 6.20996380136793041156757887995, 6.72702583523798218996067462243, 7.11612916915688730116524632128, 7.68760088927311613785086921017, 7.990453761305537290238286978106
