| L(s) = 1 | + 2·3-s + 3·4-s − 2·5-s + 3·9-s + 6·12-s − 4·15-s + 5·16-s − 6·20-s − 4·23-s + 3·25-s + 4·27-s + 4·31-s + 9·36-s − 6·45-s + 4·47-s + 10·48-s + 2·49-s + 12·59-s − 12·60-s + 3·64-s − 8·69-s + 16·71-s + 6·75-s − 10·80-s + 5·81-s − 16·89-s − 12·92-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 3/2·4-s − 0.894·5-s + 9-s + 1.73·12-s − 1.03·15-s + 5/4·16-s − 1.34·20-s − 0.834·23-s + 3/5·25-s + 0.769·27-s + 0.718·31-s + 3/2·36-s − 0.894·45-s + 0.583·47-s + 1.44·48-s + 2/7·49-s + 1.56·59-s − 1.54·60-s + 3/8·64-s − 0.963·69-s + 1.89·71-s + 0.692·75-s − 1.11·80-s + 5/9·81-s − 1.69·89-s − 1.25·92-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.882011752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.882011752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.60339523399861478911591257908, −7.10745461134828532312351558431, −6.83320522535778225263373405839, −6.45998887159628483454301412399, −6.02765959791722069150219056110, −5.40264046224584189961588499389, −4.99830573435640446190598710763, −4.20701248329351737922407289500, −4.01188876125749509547739913232, −3.52159812154109086894980433844, −2.95272499829946031906496696151, −2.59737853010778776210630568898, −2.13577887277830332901180521070, −1.57129750946872648039280326166, −0.74787223880318507666687220102,
0.74787223880318507666687220102, 1.57129750946872648039280326166, 2.13577887277830332901180521070, 2.59737853010778776210630568898, 2.95272499829946031906496696151, 3.52159812154109086894980433844, 4.01188876125749509547739913232, 4.20701248329351737922407289500, 4.99830573435640446190598710763, 5.40264046224584189961588499389, 6.02765959791722069150219056110, 6.45998887159628483454301412399, 6.83320522535778225263373405839, 7.10745461134828532312351558431, 7.60339523399861478911591257908
