L(s) = 1 | + 3·3-s + 6·9-s − 7·11-s + 2·17-s + 8·19-s + 7·25-s + 9·27-s − 21·33-s − 11·41-s + 43-s − 5·49-s + 6·51-s + 24·57-s − 59-s + 7·67-s + 14·73-s + 21·75-s + 9·81-s − 6·83-s + 12·89-s − 3·97-s − 42·99-s + 32·107-s − 14·113-s + 15·121-s − 33·123-s + 127-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 2·9-s − 2.11·11-s + 0.485·17-s + 1.83·19-s + 7/5·25-s + 1.73·27-s − 3.65·33-s − 1.71·41-s + 0.152·43-s − 5/7·49-s + 0.840·51-s + 3.17·57-s − 0.130·59-s + 0.855·67-s + 1.63·73-s + 2.42·75-s + 81-s − 0.658·83-s + 1.27·89-s − 0.304·97-s − 4.22·99-s + 3.09·107-s − 1.31·113-s + 1.36·121-s − 2.97·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.021490067\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.021490067\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 63 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−7.899030520394456735215973571712, −7.73566421574064553201839388534, −7.29241998862685358862473450646, −6.91296454232352804391096124581, −6.38540293133911611565558640611, −5.55617299690309204548939386447, −5.20778286759431769565513605570, −4.92897503593597242673462858950, −4.33866880353343236915641708850, −3.45279582029743327474111645990, −3.27746691205870178857374565883, −2.88816574598995870812108728148, −2.32175335116065284967742395307, −1.71692284008992558140237984995, −0.799693191645555365371420429636,
0.799693191645555365371420429636, 1.71692284008992558140237984995, 2.32175335116065284967742395307, 2.88816574598995870812108728148, 3.27746691205870178857374565883, 3.45279582029743327474111645990, 4.33866880353343236915641708850, 4.92897503593597242673462858950, 5.20778286759431769565513605570, 5.55617299690309204548939386447, 6.38540293133911611565558640611, 6.91296454232352804391096124581, 7.29241998862685358862473450646, 7.73566421574064553201839388534, 7.899030520394456735215973571712