L(s) = 1 | + 3·4-s + 6·7-s − 28·13-s − 7·16-s + 2·19-s + 18·28-s − 42·31-s + 106·37-s − 46·43-s − 71·49-s − 84·52-s − 142·61-s − 69·64-s − 12·67-s + 14·73-s + 6·76-s + 102·79-s − 168·91-s + 286·97-s − 386·103-s + 322·109-s − 42·112-s + 222·121-s − 126·124-s + 127-s + 131-s + 12·133-s + ⋯ |
L(s) = 1 | + 3/4·4-s + 6/7·7-s − 2.15·13-s − 0.437·16-s + 2/19·19-s + 9/14·28-s − 1.35·31-s + 2.86·37-s − 1.06·43-s − 1.44·49-s − 1.61·52-s − 2.32·61-s − 1.07·64-s − 0.179·67-s + 0.191·73-s + 3/38·76-s + 1.29·79-s − 1.84·91-s + 2.94·97-s − 3.74·103-s + 2.95·109-s − 3/8·112-s + 1.83·121-s − 1.01·124-s + 0.00787·127-s + 0.00763·131-s + 0.0902·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.085298283\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.085298283\) |
\(L(2)\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 222 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 498 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 562 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 318 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 21 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 53 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2862 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 23 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 1362 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3198 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 258 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 71 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6702 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 51 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 4222 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1262 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 143 T + p^{2} T^{2} )^{2} \) |
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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
−10.84765094349573816357212303845, −9.969055135088156734181315424229, −9.621123621556522837950332225906, −9.411970097632457632200310834353, −8.820808290966484546534507290646, −8.192985890107245584972277370265, −7.76617594573990540446715288679, −7.39835802381566846243887199120, −7.25467687474990038985954903615, −6.35553042658080005063994895899, −6.29202638112779991805864777970, −5.46546180992014601835556036027, −4.92673141531797872894234753372, −4.66857377463115535058490330635, −4.15328317732875621511305572573, −3.16601303820475977722836849564, −2.77647651815720253439690392081, −2.03001986328359106768059245269, −1.73093782297716207134725857736, −0.48882213720974476959149839870,
0.48882213720974476959149839870, 1.73093782297716207134725857736, 2.03001986328359106768059245269, 2.77647651815720253439690392081, 3.16601303820475977722836849564, 4.15328317732875621511305572573, 4.66857377463115535058490330635, 4.92673141531797872894234753372, 5.46546180992014601835556036027, 6.29202638112779991805864777970, 6.35553042658080005063994895899, 7.25467687474990038985954903615, 7.39835802381566846243887199120, 7.76617594573990540446715288679, 8.192985890107245584972277370265, 8.820808290966484546534507290646, 9.411970097632457632200310834353, 9.621123621556522837950332225906, 9.969055135088156734181315424229, 10.84765094349573816357212303845