| L(s) = 1 | − 4·9-s + 28·11-s − 280·23-s + 142·25-s − 572·29-s − 76·37-s + 68·43-s − 148·53-s − 1.36e3·67-s − 1.17e3·71-s − 2.44e3·79-s − 713·81-s − 112·99-s + 3.36e3·107-s − 1.63e3·109-s − 1.08e3·113-s − 2.07e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.80e3·169-s + ⋯ |
| L(s) = 1 | − 0.148·9-s + 0.767·11-s − 2.53·23-s + 1.13·25-s − 3.66·29-s − 0.337·37-s + 0.241·43-s − 0.383·53-s − 2.49·67-s − 1.96·71-s − 3.47·79-s − 0.978·81-s − 0.113·99-s + 3.04·107-s − 1.43·109-s − 0.899·113-s − 1.55·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.820·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 142 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1802 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9824 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 13716 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 140 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 286 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 50870 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 38 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 122000 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 66154 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 74 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 222260 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 453762 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 684 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 588 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 705072 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1220 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 964772 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1028000 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 375456 T^{2} + p^{6} T^{4} \) |
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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
−9.684022984397692245454261874020, −9.128411349068611003610056321677, −8.968678444054366104727739920644, −8.527507421274420929489114186294, −7.88210339607289002893571701690, −7.43646476898576017409533630218, −7.30859896895116771811050638705, −6.58593248224015170219759449413, −6.07977814132360318667575791600, −5.67138315321316035460112516126, −5.48367601410815890892870391696, −4.46276853386934438296346152520, −4.25213160156057497604573999231, −3.71063663694475051529453530851, −3.17908902205713486983845163468, −2.47013692941292064025370315172, −1.65669549311643921673767257976, −1.48133480202169202166090406584, 0, 0,
1.48133480202169202166090406584, 1.65669549311643921673767257976, 2.47013692941292064025370315172, 3.17908902205713486983845163468, 3.71063663694475051529453530851, 4.25213160156057497604573999231, 4.46276853386934438296346152520, 5.48367601410815890892870391696, 5.67138315321316035460112516126, 6.07977814132360318667575791600, 6.58593248224015170219759449413, 7.30859896895116771811050638705, 7.43646476898576017409533630218, 7.88210339607289002893571701690, 8.527507421274420929489114186294, 8.968678444054366104727739920644, 9.128411349068611003610056321677, 9.684022984397692245454261874020
