| L(s) = 1 | − 9·9-s − 144·11-s + 104·19-s + 156·29-s − 240·31-s + 724·41-s + 670·49-s + 1.39e3·59-s + 444·61-s − 192·71-s − 1.26e3·79-s + 81·81-s − 1.98e3·89-s + 1.29e3·99-s + 1.78e3·101-s − 892·109-s + 1.28e4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.35e3·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 3.94·11-s + 1.25·19-s + 0.998·29-s − 1.39·31-s + 2.75·41-s + 1.95·49-s + 3.07·59-s + 0.931·61-s − 0.320·71-s − 1.80·79-s + 1/9·81-s − 2.36·89-s + 1.31·99-s + 1.75·101-s − 0.783·109-s + 9.68·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.98·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.264293366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.264293366\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 670 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4358 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8382 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1230 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 78 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 78806 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 362 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 75242 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 129246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 151146 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 696 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 222 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 601510 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 96 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 746350 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 p T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 769030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 994 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 844610 T^{2} + p^{6} T^{4} \) |
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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
−9.609066596854010172545528747345, −9.255258350569777277372392849833, −8.533539371017494799688240676265, −8.349193388379483157296106515950, −7.953878536801475725799277886176, −7.53426842894149902147168880608, −7.14622652905843389522028824439, −7.03325660006936345752057073369, −5.85056114457770477083531204156, −5.70344186151030391049855193375, −5.38922975006984331840333063816, −5.14926164563545981143445552410, −4.40758048122428180016047963412, −4.02672595316898702116303425886, −3.03503269413336474233032630079, −2.90721941915083906025188167616, −2.43971861604042363296237140971, −1.97112195424329039407859023673, −0.61428967038398093731425600990, −0.60206523867416818480962169817,
0.60206523867416818480962169817, 0.61428967038398093731425600990, 1.97112195424329039407859023673, 2.43971861604042363296237140971, 2.90721941915083906025188167616, 3.03503269413336474233032630079, 4.02672595316898702116303425886, 4.40758048122428180016047963412, 5.14926164563545981143445552410, 5.38922975006984331840333063816, 5.70344186151030391049855193375, 5.85056114457770477083531204156, 7.03325660006936345752057073369, 7.14622652905843389522028824439, 7.53426842894149902147168880608, 7.953878536801475725799277886176, 8.349193388379483157296106515950, 8.533539371017494799688240676265, 9.255258350569777277372392849833, 9.609066596854010172545528747345
