| L(s) = 1 | + 6·4-s + 14·5-s + 20·16-s + 84·20-s + 18·23-s + 97·25-s + 98·31-s + 34·37-s − 64·47-s + 48·49-s − 32·53-s + 142·59-s + 24·64-s − 62·67-s + 146·71-s + 280·80-s + 18·89-s + 108·92-s − 34·97-s + 582·100-s − 32·103-s − 130·113-s + 252·115-s + 588·124-s + 322·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 14/5·5-s + 5/4·16-s + 21/5·20-s + 0.782·23-s + 3.87·25-s + 3.16·31-s + 0.918·37-s − 1.36·47-s + 0.979·49-s − 0.603·53-s + 2.40·59-s + 3/8·64-s − 0.925·67-s + 2.05·71-s + 7/2·80-s + 0.202·89-s + 1.17·92-s − 0.350·97-s + 5.81·100-s − 0.310·103-s − 1.15·113-s + 2.19·115-s + 4.74·124-s + 2.57·125-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(10.06441244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.06441244\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T^{2} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 560 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1170 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 49 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 17 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 1520 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 32 T + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 71 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 7314 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 31 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 73 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 9090 T^{2} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 12160 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12528 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
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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
−10.01524948606186479175692658378, −9.558296022145191707193031347545, −9.256209726012070296190809186602, −8.757519220548791085532378309870, −8.137259849319320112878156432762, −7.907600424657050321692870910694, −7.14840813063617745062015911594, −6.67810767473185864332906155709, −6.40052770563603359970333867415, −6.33578237492520727696628073616, −5.66315834758724107405792899010, −5.40518828495700970659082325800, −4.89536993201924350298564260728, −4.30410947364597596732599470767, −3.35441212460530370623727187718, −2.77278635843722614241562168013, −2.28336081677831405515085277785, −2.28305346052858889459469012778, −1.29623071465634502797093564536, −1.07775334221641359095138732183,
1.07775334221641359095138732183, 1.29623071465634502797093564536, 2.28305346052858889459469012778, 2.28336081677831405515085277785, 2.77278635843722614241562168013, 3.35441212460530370623727187718, 4.30410947364597596732599470767, 4.89536993201924350298564260728, 5.40518828495700970659082325800, 5.66315834758724107405792899010, 6.33578237492520727696628073616, 6.40052770563603359970333867415, 6.67810767473185864332906155709, 7.14840813063617745062015911594, 7.907600424657050321692870910694, 8.137259849319320112878156432762, 8.757519220548791085532378309870, 9.256209726012070296190809186602, 9.558296022145191707193031347545, 10.01524948606186479175692658378
