| L(s) = 1 | + 8·11-s + 10·19-s + 4·29-s + 14·31-s + 10·49-s − 28·59-s − 22·61-s + 6·79-s + 24·101-s + 38·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 2.29·19-s + 0.742·29-s + 2.51·31-s + 10/7·49-s − 3.64·59-s − 2.81·61-s + 0.675·79-s + 2.38·101-s + 3.63·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.846376746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.846376746\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 165 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.433019509621554621510337574800, −7.76350846739432601511761921944, −7.59832217774317591493071614243, −7.50967502341786347062760270551, −6.76036459566797312965647007126, −6.57721799896623568344948336535, −6.16725778531207470312957506781, −6.06788941703415977485679885864, −5.54095793334687887579363663852, −4.93462674203343226504718087974, −4.55130856256476425388310426825, −4.50001122267334841834769597219, −3.88678061193313912068876285760, −3.42006851196004298677118705664, −3.00032659809064789519806859838, −2.90374307619972749451413630349, −1.99195114414360831362003479552, −1.46371370780795108627137448942, −1.10339792779453947612749867421, −0.69059870266953878768696197914,
0.69059870266953878768696197914, 1.10339792779453947612749867421, 1.46371370780795108627137448942, 1.99195114414360831362003479552, 2.90374307619972749451413630349, 3.00032659809064789519806859838, 3.42006851196004298677118705664, 3.88678061193313912068876285760, 4.50001122267334841834769597219, 4.55130856256476425388310426825, 4.93462674203343226504718087974, 5.54095793334687887579363663852, 6.06788941703415977485679885864, 6.16725778531207470312957506781, 6.57721799896623568344948336535, 6.76036459566797312965647007126, 7.50967502341786347062760270551, 7.59832217774317591493071614243, 7.76350846739432601511761921944, 8.433019509621554621510337574800
