| L(s) = 1 | + 7-s + 2·9-s − 13-s − 17-s + 19-s + 23-s + 2·25-s − 29-s − 41-s + 59-s − 61-s + 2·63-s + 79-s + 3·81-s + 83-s − 91-s − 97-s − 2·103-s − 4·107-s − 2·117-s − 119-s + 2·121-s + 127-s + 131-s + 133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 7-s + 2·9-s − 13-s − 17-s + 19-s + 23-s + 2·25-s − 29-s − 41-s + 59-s − 61-s + 2·63-s + 79-s + 3·81-s + 83-s − 91-s − 97-s − 2·103-s − 4·107-s − 2·117-s − 119-s + 2·121-s + 127-s + 131-s + 133-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2715904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2715904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.576541903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.576541903\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 103 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 29 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 61 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 83 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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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
−9.757403664164021652333957560855, −9.426703236526489620931608739654, −8.953757265651555873030935734700, −8.718330890008259352810445859702, −8.024985188331377061357889699760, −7.77987749194725253248290054761, −7.23672668661631675728880311503, −7.13120967147530434256920310349, −6.62413043478103930475680738820, −6.43995842729002153357305259811, −5.35337503433118517222732619290, −5.18750082606978775067646190720, −4.85348678406918890477298429116, −4.47875948076031876896739814694, −3.96875473760134105011147445352, −3.47249176095633824252218411745, −2.73165974518830779494489174995, −2.27451909750472764727401528162, −1.47297080129859474232324611776, −1.20267237865185685088978427625,
1.20267237865185685088978427625, 1.47297080129859474232324611776, 2.27451909750472764727401528162, 2.73165974518830779494489174995, 3.47249176095633824252218411745, 3.96875473760134105011147445352, 4.47875948076031876896739814694, 4.85348678406918890477298429116, 5.18750082606978775067646190720, 5.35337503433118517222732619290, 6.43995842729002153357305259811, 6.62413043478103930475680738820, 7.13120967147530434256920310349, 7.23672668661631675728880311503, 7.77987749194725253248290054761, 8.024985188331377061357889699760, 8.718330890008259352810445859702, 8.953757265651555873030935734700, 9.426703236526489620931608739654, 9.757403664164021652333957560855
