| L(s) = 1 | + 2·2-s + 2·4-s + 2·7-s + 5·9-s + 4·14-s − 4·16-s + 10·18-s − 2·23-s + 9·25-s + 4·28-s − 8·32-s + 10·36-s − 4·46-s − 3·49-s + 18·50-s + 10·63-s − 8·64-s + 6·71-s + 20·79-s + 16·81-s − 4·92-s − 6·98-s + 18·100-s − 8·112-s − 18·113-s + 121-s + 20·126-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.755·7-s + 5/3·9-s + 1.06·14-s − 16-s + 2.35·18-s − 0.417·23-s + 9/5·25-s + 0.755·28-s − 1.41·32-s + 5/3·36-s − 0.589·46-s − 3/7·49-s + 2.54·50-s + 1.25·63-s − 64-s + 0.712·71-s + 2.25·79-s + 16/9·81-s − 0.417·92-s − 0.606·98-s + 9/5·100-s − 0.755·112-s − 1.69·113-s + 1/11·121-s + 1.78·126-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379456 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.948458482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.948458482\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.659513154772004960747939965579, −8.070586489854203231719899307720, −7.69094210516922989048916236028, −7.08400975227045809813806739185, −6.71488663354558602008507846885, −6.41389865234774982366788610714, −5.68147398799127885593893561203, −5.14925595534656207307949351424, −4.71843472652079736037939416635, −4.48121952748642591535418801717, −3.81429290402862768774153946508, −3.39208462043531436058498962670, −2.56614151422556875497636638904, −1.91728484706753626802777118462, −1.09772374240400117663505626821,
1.09772374240400117663505626821, 1.91728484706753626802777118462, 2.56614151422556875497636638904, 3.39208462043531436058498962670, 3.81429290402862768774153946508, 4.48121952748642591535418801717, 4.71843472652079736037939416635, 5.14925595534656207307949351424, 5.68147398799127885593893561203, 6.41389865234774982366788610714, 6.71488663354558602008507846885, 7.08400975227045809813806739185, 7.69094210516922989048916236028, 8.070586489854203231719899307720, 8.659513154772004960747939965579
