| L(s) = 1 | + 8·3-s − 4·5-s + 32·9-s + 12·11-s − 32·15-s − 16-s + 2·25-s + 88·27-s + 8·31-s + 96·33-s − 16·37-s − 128·45-s + 20·47-s − 8·48-s − 24·53-s − 48·55-s + 12·67-s + 32·71-s + 16·75-s + 4·80-s + 206·81-s + 64·93-s + 8·97-s + 384·99-s + 12·103-s − 128·111-s + 52·113-s + ⋯ |
| L(s) = 1 | + 4.61·3-s − 1.78·5-s + 32/3·9-s + 3.61·11-s − 8.26·15-s − 1/4·16-s + 2/5·25-s + 16.9·27-s + 1.43·31-s + 16.7·33-s − 2.63·37-s − 19.0·45-s + 2.91·47-s − 1.15·48-s − 3.29·53-s − 6.47·55-s + 1.46·67-s + 3.79·71-s + 1.84·75-s + 0.447·80-s + 22.8·81-s + 6.63·93-s + 0.812·97-s + 38.5·99-s + 1.18·103-s − 12.1·111-s + 4.89·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.86836655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.86836655\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 64 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 3214 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 96 T^{2} + p^{2} T^{4} )( 1 + 96 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 108 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 9422 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.48263829700447805087406912619, −7.42228017478961630488691247979, −7.23048649971555986425190608223, −6.90199617582853896403059175197, −6.59751100078320851016459118319, −6.42467303392393843388064657247, −6.31500948553918222672058831076, −6.02128331918261053954021301463, −5.42890308659550983019848431708, −5.04858323021295868530515840936, −4.66292806403760146006033867634, −4.56883112588492768511340834689, −4.20002169759427169217719176419, −4.00941523822253993473325558394, −3.80020024167252191060552306712, −3.49152382259985417621346215858, −3.48389808131150981733814786214, −3.26849221659852822896550991430, −3.12792976410519124061027458445, −2.56700738414889156815272863635, −2.13192332177002063512102761674, −2.01403252375171773576052575104, −1.89627815102061108570217478180, −1.06187127035623426419716766658, −0.931388580131252379563727441700,
0.931388580131252379563727441700, 1.06187127035623426419716766658, 1.89627815102061108570217478180, 2.01403252375171773576052575104, 2.13192332177002063512102761674, 2.56700738414889156815272863635, 3.12792976410519124061027458445, 3.26849221659852822896550991430, 3.48389808131150981733814786214, 3.49152382259985417621346215858, 3.80020024167252191060552306712, 4.00941523822253993473325558394, 4.20002169759427169217719176419, 4.56883112588492768511340834689, 4.66292806403760146006033867634, 5.04858323021295868530515840936, 5.42890308659550983019848431708, 6.02128331918261053954021301463, 6.31500948553918222672058831076, 6.42467303392393843388064657247, 6.59751100078320851016459118319, 6.90199617582853896403059175197, 7.23048649971555986425190608223, 7.42228017478961630488691247979, 7.48263829700447805087406912619
