| L(s) = 1 | + 4·2-s + 2·3-s + 10·4-s + 8·6-s − 6·7-s + 20·8-s + 2·9-s + 20·12-s + 4·13-s − 24·14-s + 35·16-s + 8·18-s − 12·21-s + 16·23-s + 40·24-s + 16·26-s + 6·27-s − 60·28-s + 56·32-s + 20·36-s + 8·39-s − 8·41-s − 48·42-s + 64·46-s + 70·48-s + 18·49-s + 40·52-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 5·4-s + 3.26·6-s − 2.26·7-s + 7.07·8-s + 2/3·9-s + 5.77·12-s + 1.10·13-s − 6.41·14-s + 35/4·16-s + 1.88·18-s − 2.61·21-s + 3.33·23-s + 8.16·24-s + 3.13·26-s + 1.15·27-s − 11.3·28-s + 9.89·32-s + 10/3·36-s + 1.28·39-s − 1.24·41-s − 7.40·42-s + 9.43·46-s + 10.1·48-s + 18/7·49-s + 5.54·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(35.83971544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(35.83971544\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 798 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6158 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 20 T^{2} - 1322 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8998 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 18 T + 222 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 19198 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 6 T + 198 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.95072063394678907984108244673, −6.82865924559308352465896084731, −6.68441058385257951394634051310, −6.41363567199401952692838310842, −6.19742712720476688517660351696, −5.68336357185931872345313926735, −5.68204920035240415023245852844, −5.62441890930335818294982644520, −5.46030595039366272031243137357, −4.81262740963840833702965431464, −4.69365316308581384196775364594, −4.60764282010960848294275498108, −4.20542923027857181654649450472, −4.02328040981591173493777339452, −3.73199676424986754950680523977, −3.36536578159568296727130980861, −3.36435073271541524964057958057, −2.93725583966169976163329291474, −2.92452096857446816300009630787, −2.78253625620746252663217630784, −2.52524051686233231605210669586, −1.82199335561223215722610355937, −1.67404563871674716494633397110, −1.14919653945494743567970880901, −0.69110077047701441249159540905,
0.69110077047701441249159540905, 1.14919653945494743567970880901, 1.67404563871674716494633397110, 1.82199335561223215722610355937, 2.52524051686233231605210669586, 2.78253625620746252663217630784, 2.92452096857446816300009630787, 2.93725583966169976163329291474, 3.36435073271541524964057958057, 3.36536578159568296727130980861, 3.73199676424986754950680523977, 4.02328040981591173493777339452, 4.20542923027857181654649450472, 4.60764282010960848294275498108, 4.69365316308581384196775364594, 4.81262740963840833702965431464, 5.46030595039366272031243137357, 5.62441890930335818294982644520, 5.68204920035240415023245852844, 5.68336357185931872345313926735, 6.19742712720476688517660351696, 6.41363567199401952692838310842, 6.68441058385257951394634051310, 6.82865924559308352465896084731, 6.95072063394678907984108244673
