| L(s) = 1 | − 4-s − 2·7-s − 4·13-s − 3·16-s + 8·19-s + 2·25-s + 2·28-s − 8·31-s + 4·37-s + 8·43-s + 3·49-s + 4·52-s + 20·61-s + 7·64-s − 8·67-s − 28·73-s − 8·76-s − 16·79-s + 8·91-s + 28·97-s − 2·100-s − 8·103-s − 4·109-s + 6·112-s + 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.755·7-s − 1.10·13-s − 3/4·16-s + 1.83·19-s + 2/5·25-s + 0.377·28-s − 1.43·31-s + 0.657·37-s + 1.21·43-s + 3/7·49-s + 0.554·52-s + 2.56·61-s + 7/8·64-s − 0.977·67-s − 3.27·73-s − 0.917·76-s − 1.80·79-s + 0.838·91-s + 2.84·97-s − 1/5·100-s − 0.788·103-s − 0.383·109-s + 0.566·112-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58110129 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8217605907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8217605907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−7.966166803274839891296484269086, −7.59264416772538769030529440664, −7.23500657013727562323978795066, −7.08741353505528136254336997061, −6.93905191034998487611484152844, −6.17996084228759277528875838951, −5.94591917222670399098616490672, −5.58078473018795340354153216957, −5.28931211461111501227249077658, −4.87438668852963856926831226846, −4.44427879050057226096831479175, −4.27079603230297123164920223355, −3.54638181272209934611027705494, −3.49731789265229192892405087545, −2.81221919345149224655959512512, −2.59713974653824407903759550558, −2.14733469973242328850796662666, −1.44043432488147135349688931945, −0.934852716066425773251603952266, −0.25790234132819762066955397447,
0.25790234132819762066955397447, 0.934852716066425773251603952266, 1.44043432488147135349688931945, 2.14733469973242328850796662666, 2.59713974653824407903759550558, 2.81221919345149224655959512512, 3.49731789265229192892405087545, 3.54638181272209934611027705494, 4.27079603230297123164920223355, 4.44427879050057226096831479175, 4.87438668852963856926831226846, 5.28931211461111501227249077658, 5.58078473018795340354153216957, 5.94591917222670399098616490672, 6.17996084228759277528875838951, 6.93905191034998487611484152844, 7.08741353505528136254336997061, 7.23500657013727562323978795066, 7.59264416772538769030529440664, 7.966166803274839891296484269086
