| L(s) = 1 | − 4·7-s − 8·9-s − 8·17-s + 8·23-s − 16·25-s + 16·31-s + 8·41-s + 16·47-s + 10·49-s + 32·63-s + 8·71-s − 8·73-s + 24·79-s + 32·81-s + 24·89-s − 8·97-s + 32·103-s + 16·113-s + 32·119-s − 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 64·153-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 8/3·9-s − 1.94·17-s + 1.66·23-s − 3.19·25-s + 2.87·31-s + 1.24·41-s + 2.33·47-s + 10/7·49-s + 4.03·63-s + 0.949·71-s − 0.936·73-s + 2.70·79-s + 32/9·81-s + 2.54·89-s − 0.812·97-s + 3.15·103-s + 1.50·113-s + 2.93·119-s − 2.54·121-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 + 5.17·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.708911414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708911414\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 + 16 T^{2} + 112 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 28 T^{2} + 406 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 16 T^{2} + 240 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 56 T^{2} + 1504 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 36 T^{2} + 438 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 68 T^{2} + 2326 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2:C_4$ | \( 1 + 156 T^{2} + 9750 T^{4} + 156 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 148 T^{2} + 10582 T^{4} + 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 104 T^{2} + 6304 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 240 T^{2} + 21840 T^{4} + 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 - 20 T^{2} - 1290 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 12 T + 186 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 296 T^{2} + 35520 T^{4} + 296 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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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
−6.23813844552869276319573079060, −5.75440910098324942306923955755, −5.70278615150439147218185227676, −5.57240800881527155363625541595, −5.51725979299595917448592402412, −4.86333128488602900128200134918, −4.84590583227539973970444193775, −4.78192156948990625210866106320, −4.45400140891666091161612574545, −4.17601811482592955979872554733, −3.87922650391284749474284875675, −3.80559607003659133085286913051, −3.67515084204233894244841416569, −3.05649471415000043803589348362, −3.04640810099335768491267406323, −2.96749087197729437968140914994, −2.86480082712634735612874278108, −2.29564805135326596811201423809, −2.13737627105708479104635022905, −2.05569721887651587511915269765, −2.01495096706113076394506257075, −0.981824710816325047550763913564, −0.879003626013735253258174430769, −0.50573005216320622909249495709, −0.33216338296864354427139536722,
0.33216338296864354427139536722, 0.50573005216320622909249495709, 0.879003626013735253258174430769, 0.981824710816325047550763913564, 2.01495096706113076394506257075, 2.05569721887651587511915269765, 2.13737627105708479104635022905, 2.29564805135326596811201423809, 2.86480082712634735612874278108, 2.96749087197729437968140914994, 3.04640810099335768491267406323, 3.05649471415000043803589348362, 3.67515084204233894244841416569, 3.80559607003659133085286913051, 3.87922650391284749474284875675, 4.17601811482592955979872554733, 4.45400140891666091161612574545, 4.78192156948990625210866106320, 4.84590583227539973970444193775, 4.86333128488602900128200134918, 5.51725979299595917448592402412, 5.57240800881527155363625541595, 5.70278615150439147218185227676, 5.75440910098324942306923955755, 6.23813844552869276319573079060
