| L(s) = 1 | − 3-s − 4-s − 5-s − 5·7-s + 9-s + 12-s + 15-s + 16-s + 20-s + 5·21-s − 4·25-s − 27-s + 5·28-s + 5·35-s − 36-s − 5·37-s − 10·41-s − 15·43-s − 45-s + 47-s − 48-s + 18·49-s + 20·59-s − 60-s − 5·63-s − 64-s + 4·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s − 0.447·5-s − 1.88·7-s + 1/3·9-s + 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.223·20-s + 1.09·21-s − 4/5·25-s − 0.192·27-s + 0.944·28-s + 0.845·35-s − 1/6·36-s − 0.821·37-s − 1.56·41-s − 2.28·43-s − 0.149·45-s + 0.145·47-s − 0.144·48-s + 18/7·49-s + 2.60·59-s − 0.129·60-s − 0.629·63-s − 1/8·64-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132300 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4379027794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4379027794\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 79 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.607208471564681200929939698031, −8.848816285099318986087744849626, −8.505698999019485589977521256618, −7.984567411096763444580986362622, −7.13682700831533312015440131557, −6.94316955660584393926016610480, −6.41578453280906087258831566326, −5.88874387429572538588576714330, −5.30963037576648192115197073588, −4.81816101464795294409726127939, −3.91421469398545160810260207237, −3.60792296384436066119156884197, −3.03950771406390117947679780329, −1.93991427194705757404501644184, −0.46134343672254373040015387019,
0.46134343672254373040015387019, 1.93991427194705757404501644184, 3.03950771406390117947679780329, 3.60792296384436066119156884197, 3.91421469398545160810260207237, 4.81816101464795294409726127939, 5.30963037576648192115197073588, 5.88874387429572538588576714330, 6.41578453280906087258831566326, 6.94316955660584393926016610480, 7.13682700831533312015440131557, 7.984567411096763444580986362622, 8.505698999019485589977521256618, 8.848816285099318986087744849626, 9.607208471564681200929939698031
