| L(s) = 1 | + 4-s + 5-s − 2·9-s − 3·16-s + 12·19-s + 20-s + 25-s + 12·29-s − 4·31-s − 2·36-s − 12·41-s − 2·45-s − 14·49-s − 12·59-s − 12·61-s − 7·64-s + 12·76-s − 12·79-s − 3·80-s − 5·81-s + 12·89-s + 12·95-s + 100-s + 24·101-s + 12·116-s − 11·121-s − 4·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.447·5-s − 2/3·9-s − 3/4·16-s + 2.75·19-s + 0.223·20-s + 1/5·25-s + 2.22·29-s − 0.718·31-s − 1/3·36-s − 1.87·41-s − 0.298·45-s − 2·49-s − 1.56·59-s − 1.53·61-s − 7/8·64-s + 1.37·76-s − 1.35·79-s − 0.335·80-s − 5/9·81-s + 1.27·89-s + 1.23·95-s + 1/10·100-s + 2.38·101-s + 1.11·116-s − 121-s − 0.359·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.260735718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.260735718\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 - T \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
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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
−11.30107201495925650682043592330, −10.49991131790129820178486205353, −10.11609106150667627357871518841, −9.442328118663931367009636881169, −9.045465813848817190841343143300, −8.346862351224876521059923355288, −7.70472128585836812440011641667, −7.13649330025224367787606250318, −6.48225542860650127281451265108, −5.96114101219285936909826337141, −5.10880251903835079367118589419, −4.73150026167311415697709554733, −3.25393235136405758234082174005, −2.93126651922222444824982053212, −1.57895640535438383210394971746,
1.57895640535438383210394971746, 2.93126651922222444824982053212, 3.25393235136405758234082174005, 4.73150026167311415697709554733, 5.10880251903835079367118589419, 5.96114101219285936909826337141, 6.48225542860650127281451265108, 7.13649330025224367787606250318, 7.70472128585836812440011641667, 8.346862351224876521059923355288, 9.045465813848817190841343143300, 9.442328118663931367009636881169, 10.11609106150667627357871518841, 10.49991131790129820178486205353, 11.30107201495925650682043592330
