| L(s) = 1 | + 16·25-s + 28·49-s + 64·73-s − 32·97-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 16/5·25-s + 4·49-s + 7.49·73-s − 3.24·97-s + 4·121-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 − 1.53·169-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 + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.548382744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.548382744\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12200634456221861534024840262, −6.80383903855519945695903903644, −6.68746950410165444279736717494, −6.29559020225044511567527529003, −6.23345177835311418834626686350, −5.88555992163844546460120828132, −5.58961138750239238582698113196, −5.43353300245555024798488117957, −5.21615715182982519759331240706, −4.92632111829821756938065375860, −4.82224861148872943004626448403, −4.50557470046577056310655699608, −4.33309291259683086799122636166, −3.72521529742037320006668361241, −3.72425660230411444263024043727, −3.71157720122487796530296802852, −3.14220198257125770351731104344, −2.90909340546858228929072609915, −2.48507566557635206725054555884, −2.41563130902469822184928660742, −2.17109323394852428493266134422, −1.66790207971258112415890313904, −1.06128194526495515645448028093, −0.904717735467217663469192575429, −0.58578406914842825321231486361,
0.58578406914842825321231486361, 0.904717735467217663469192575429, 1.06128194526495515645448028093, 1.66790207971258112415890313904, 2.17109323394852428493266134422, 2.41563130902469822184928660742, 2.48507566557635206725054555884, 2.90909340546858228929072609915, 3.14220198257125770351731104344, 3.71157720122487796530296802852, 3.72425660230411444263024043727, 3.72521529742037320006668361241, 4.33309291259683086799122636166, 4.50557470046577056310655699608, 4.82224861148872943004626448403, 4.92632111829821756938065375860, 5.21615715182982519759331240706, 5.43353300245555024798488117957, 5.58961138750239238582698113196, 5.88555992163844546460120828132, 6.23345177835311418834626686350, 6.29559020225044511567527529003, 6.68746950410165444279736717494, 6.80383903855519945695903903644, 7.12200634456221861534024840262
