L(s) = 1 | − 4·5-s + 8·7-s + 4·9-s − 8·11-s − 12·13-s − 4·25-s − 32·35-s + 24·43-s − 16·45-s + 34·49-s + 32·55-s + 12·61-s + 32·63-s + 48·65-s − 24·67-s − 64·77-s + 6·81-s − 96·91-s − 32·99-s + 4·101-s + 48·103-s + 8·107-s − 24·113-s − 48·117-s + 20·121-s + 44·125-s + 127-s + ⋯ |
L(s) = 1 | − 1.78·5-s + 3.02·7-s + 4/3·9-s − 2.41·11-s − 3.32·13-s − 4/5·25-s − 5.40·35-s + 3.65·43-s − 2.38·45-s + 34/7·49-s + 4.31·55-s + 1.53·61-s + 4.03·63-s + 5.95·65-s − 2.93·67-s − 7.29·77-s + 2/3·81-s − 10.0·91-s − 3.21·99-s + 0.398·101-s + 4.72·103-s + 0.773·107-s − 2.25·113-s − 4.43·117-s + 1.81·121-s + 3.93·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.737250394\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.737250394\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
good | 3 | $D_4\times C_2$ | \( 1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 1686 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 19530 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.38232468742527731645632474376, −7.23175189388237959629776825590, −7.22012803478872992568963484150, −6.78954321362138876887089126078, −6.40750983352129277576636035264, −5.93077526664128178923172511820, −5.70067820394470215721721253687, −5.42861085727568203242823398289, −5.41316743239096358681565202596, −5.12448716454450108178972077456, −4.72819708528956310774108383687, −4.63742678297712441093110923420, −4.46107299867152544962790831716, −4.23032418459533185595911699425, −4.21856786524656515731551904812, −3.84777846204173715390377319389, −3.18364242530950448584662721867, −3.10497425902761634594239660906, −2.62089834895025186585259445390, −2.30632300220758869695322856539, −2.07947298246006322917760616113, −1.97771810361146053890213385609, −1.50965111345812625728243321209, −0.66277459923089049582352058904, −0.45624222994416594933553733501,
0.45624222994416594933553733501, 0.66277459923089049582352058904, 1.50965111345812625728243321209, 1.97771810361146053890213385609, 2.07947298246006322917760616113, 2.30632300220758869695322856539, 2.62089834895025186585259445390, 3.10497425902761634594239660906, 3.18364242530950448584662721867, 3.84777846204173715390377319389, 4.21856786524656515731551904812, 4.23032418459533185595911699425, 4.46107299867152544962790831716, 4.63742678297712441093110923420, 4.72819708528956310774108383687, 5.12448716454450108178972077456, 5.41316743239096358681565202596, 5.42861085727568203242823398289, 5.70067820394470215721721253687, 5.93077526664128178923172511820, 6.40750983352129277576636035264, 6.78954321362138876887089126078, 7.22012803478872992568963484150, 7.23175189388237959629776825590, 7.38232468742527731645632474376