| L(s) = 1 | − 5-s − 22·11-s − 35·23-s + 25·25-s − 37·31-s + 25·37-s + 100·47-s + 98·49-s + 140·53-s + 22·55-s − 107·59-s + 35·67-s + 133·71-s − 97·89-s − 95·97-s + 380·103-s + 215·113-s + 35·115-s + 363·121-s − 74·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 37·155-s + ⋯ |
| L(s) = 1 | − 1/5·5-s − 2·11-s − 1.52·23-s + 25-s − 1.19·31-s + 0.675·37-s + 2.12·47-s + 2·49-s + 2.64·53-s + 2/5·55-s − 1.81·59-s + 0.522·67-s + 1.87·71-s − 1.08·89-s − 0.979·97-s + 3.68·103-s + 1.90·113-s + 7/23·115-s + 3·121-s − 0.591·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.238·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2509056 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.678070282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.678070282\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 24 T^{2} + p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T + 696 T^{2} + 35 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 37 T + 408 T^{2} + 37 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T - 744 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 107 T + 7968 T^{2} + 107 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 35 T - 3264 T^{2} - 35 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 133 T + 12648 T^{2} - 133 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 97 T + 1488 T^{2} + 97 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 95 T - 384 T^{2} + 95 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−9.379925407138494022336187500733, −8.979141358046744401718523136416, −8.546743388893875586090151511044, −8.352866878490961205059824350128, −7.63546658731444965585532985166, −7.53655142305454357701561520426, −7.24633686536194978258263392308, −6.68666641823714000335451223329, −5.91071065390175569022347336533, −5.85791914706453227652528984263, −5.32961407865246960548285954206, −4.99372139788301903002411934845, −4.33545581512937863458919565480, −4.00853351250419451658636899497, −3.46506909828205863418179698408, −2.83711206751063116636007782584, −2.23426977340438872556270924408, −2.17918240927245145478945171632, −0.954778021733146979436581372230, −0.41823043148460933934833604088,
0.41823043148460933934833604088, 0.954778021733146979436581372230, 2.17918240927245145478945171632, 2.23426977340438872556270924408, 2.83711206751063116636007782584, 3.46506909828205863418179698408, 4.00853351250419451658636899497, 4.33545581512937863458919565480, 4.99372139788301903002411934845, 5.32961407865246960548285954206, 5.85791914706453227652528984263, 5.91071065390175569022347336533, 6.68666641823714000335451223329, 7.24633686536194978258263392308, 7.53655142305454357701561520426, 7.63546658731444965585532985166, 8.352866878490961205059824350128, 8.546743388893875586090151511044, 8.979141358046744401718523136416, 9.379925407138494022336187500733
