| L(s) = 1 | − 7-s − 2·13-s + 19-s + 5·25-s − 14·31-s + 10·37-s − 5·43-s − 6·49-s − 2·61-s − 32·67-s − 17·73-s − 8·79-s + 2·91-s + 19·97-s − 20·103-s − 17·109-s + 11·121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.554·13-s + 0.229·19-s + 25-s − 2.51·31-s + 1.64·37-s − 0.762·43-s − 6/7·49-s − 0.256·61-s − 3.90·67-s − 1.98·73-s − 0.900·79-s + 0.209·91-s + 1.92·97-s − 1.97·103-s − 1.62·109-s + 121-s + 0.0887·127-s + 0.0873·131-s − 0.0867·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015417720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015417720\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−9.266767651967518653346250148897, −8.888572742096398216758273610861, −8.577333052580492898044250277417, −7.81712759934282365097757900860, −7.79981764771986420062339712429, −7.24622133768332024098581760994, −6.98526001907009710630894561713, −6.53477433935067566432476937950, −6.05482339964846725006141148652, −5.62253722790853836220016482228, −5.42116715680561625516084718843, −4.60013963629685721316710159191, −4.58874496882461498747731741089, −3.95157708761761045975569086885, −3.37661580573769109570361439991, −2.92378891417182228722388778675, −2.65683191394103980415419442736, −1.72349709382736545315948360924, −1.44810529292625391476818320843, −0.34344478169413923337686489562,
0.34344478169413923337686489562, 1.44810529292625391476818320843, 1.72349709382736545315948360924, 2.65683191394103980415419442736, 2.92378891417182228722388778675, 3.37661580573769109570361439991, 3.95157708761761045975569086885, 4.58874496882461498747731741089, 4.60013963629685721316710159191, 5.42116715680561625516084718843, 5.62253722790853836220016482228, 6.05482339964846725006141148652, 6.53477433935067566432476937950, 6.98526001907009710630894561713, 7.24622133768332024098581760994, 7.79981764771986420062339712429, 7.81712759934282365097757900860, 8.577333052580492898044250277417, 8.888572742096398216758273610861, 9.266767651967518653346250148897
