| L(s) = 1 | + 10·7-s − 14·25-s − 8·37-s + 8·43-s + 61·49-s − 8·67-s − 32·79-s − 56·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 44·169-s + 173-s − 140·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 3.77·7-s − 2.79·25-s − 1.31·37-s + 1.21·43-s + 61/7·49-s − 0.977·67-s − 3.60·79-s − 5.36·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-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 − 3.38·169-s + 0.0760·173-s − 10.5·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02728681240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02728681240\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 35 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 163 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.03214471779589205005105606008, −5.74689395077437169558879317679, −5.70071459696787904124395162795, −5.58537974474968427434630323766, −5.44684777649556931741021028492, −5.06523659664105468867235254468, −4.96970325756372227392625289919, −4.86675200679526904668666951427, −4.48751833718209563933944894384, −4.30440038727579681045027624422, −4.04490367473859350494817356454, −4.01934717521765141056227569094, −3.97247421528029054798734240895, −3.54425636158699774407137319281, −3.16859811553518843344275301406, −2.83753524379660739225850369003, −2.62734749831341312664469335812, −2.40486850884666926516643251749, −2.04604104851733677871918777163, −1.87890999684108893766373150336, −1.71898678417196848080415659740, −1.35476923484616578687094700357, −1.22565207443638167004964372409, −0.931553221629601885976245899392, −0.01989054832205751298723767179,
0.01989054832205751298723767179, 0.931553221629601885976245899392, 1.22565207443638167004964372409, 1.35476923484616578687094700357, 1.71898678417196848080415659740, 1.87890999684108893766373150336, 2.04604104851733677871918777163, 2.40486850884666926516643251749, 2.62734749831341312664469335812, 2.83753524379660739225850369003, 3.16859811553518843344275301406, 3.54425636158699774407137319281, 3.97247421528029054798734240895, 4.01934717521765141056227569094, 4.04490367473859350494817356454, 4.30440038727579681045027624422, 4.48751833718209563933944894384, 4.86675200679526904668666951427, 4.96970325756372227392625289919, 5.06523659664105468867235254468, 5.44684777649556931741021028492, 5.58537974474968427434630323766, 5.70071459696787904124395162795, 5.74689395077437169558879317679, 6.03214471779589205005105606008
