| L(s) = 1 | − 11·4-s + 4·5-s + 57·16-s − 44·20-s − 224·23-s − 238·25-s + 240·31-s + 772·37-s + 472·47-s − 186·49-s + 156·53-s − 1.68e3·59-s + 77·64-s − 552·67-s − 1.14e3·71-s + 228·80-s − 1.82e3·89-s + 2.46e3·92-s + 772·97-s + 2.61e3·100-s − 496·103-s − 4.52e3·113-s − 896·115-s − 2.64e3·124-s − 1.46e3·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.37·4-s + 0.357·5-s + 0.890·16-s − 0.491·20-s − 2.03·23-s − 1.90·25-s + 1.39·31-s + 3.43·37-s + 1.46·47-s − 0.542·49-s + 0.404·53-s − 3.70·59-s + 0.150·64-s − 1.00·67-s − 1.91·71-s + 0.318·80-s − 2.17·89-s + 2.79·92-s + 0.808·97-s + 2.61·100-s − 0.474·103-s − 3.76·113-s − 0.726·115-s − 1.91·124-s − 1.05·125-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 11 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 186 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3674 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2606 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 1218 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 15158 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 118622 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 131634 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 236 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 840 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58038 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 276 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 572 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 393246 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 441578 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 85574 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 914 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 386 T + p^{3} T^{2} )^{2} \) |
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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
−9.272862076252555844399636454108, −9.101391478189125890206917118263, −8.361122339221887355348019933531, −8.101242713726487435369747765556, −7.61491375444902652710334143115, −7.61016287514245969344163969229, −6.57499701940796608756138237250, −6.11104197897770786236581259474, −5.87628080986162841213012534638, −5.61753104155182809200811512294, −4.60983816999202034362338147024, −4.56255524496584585812361484478, −4.08345558176028529331813739510, −3.73221992305697300589207807685, −2.75438470456173966325561137421, −2.52107861494217726199710977969, −1.60017082780532440952397411072, −1.08643237081168198023980816725, 0, 0,
1.08643237081168198023980816725, 1.60017082780532440952397411072, 2.52107861494217726199710977969, 2.75438470456173966325561137421, 3.73221992305697300589207807685, 4.08345558176028529331813739510, 4.56255524496584585812361484478, 4.60983816999202034362338147024, 5.61753104155182809200811512294, 5.87628080986162841213012534638, 6.11104197897770786236581259474, 6.57499701940796608756138237250, 7.61016287514245969344163969229, 7.61491375444902652710334143115, 8.101242713726487435369747765556, 8.361122339221887355348019933531, 9.101391478189125890206917118263, 9.272862076252555844399636454108
