| L(s) = 1 | + 2·4-s + 8·9-s + 16-s − 16·25-s − 16·29-s + 16·36-s − 64·43-s − 40·53-s − 2·64-s + 8·79-s + 34·81-s − 32·100-s − 32·107-s + 96·113-s − 32·116-s − 12·121-s + 127-s + 131-s + 137-s + 139-s + 8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 48·169-s + ⋯ |
| L(s) = 1 | + 4-s + 8/3·9-s + 1/4·16-s − 3.19·25-s − 2.97·29-s + 8/3·36-s − 9.75·43-s − 5.49·53-s − 1/4·64-s + 0.900·79-s + 34/9·81-s − 3.19·100-s − 3.09·107-s + 9.03·113-s − 2.97·116-s − 1.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.301289333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.301289333\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | \( 1 \) |
| 13 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 5 | \( ( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 6 T^{2} - 85 T^{4} + 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 26 T^{2} + 387 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 20 T^{2} + 39 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 31 | \( ( 1 - 10 T^{2} - 861 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 86 T^{2} + 5187 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 68 T^{2} + 1143 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 120 T^{2} + 10679 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 118 T^{2} + 9435 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 54 T^{2} - 2413 T^{4} - 54 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 T - 75 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 164 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 106 T^{2} + 3315 T^{4} + 106 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 162 T^{2} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.18976783455498990837800475709, −4.09594833455184229815712036545, −3.67253730929176898173946710656, −3.61227013324088992179442689442, −3.53224979246201419615844734503, −3.49834041296306632400281689801, −3.31793641467848323867108434559, −3.27359972307916229728672016501, −3.16695376294254708959347273991, −3.15664164261279725242054342114, −3.00079794300709524015225920262, −2.48876129939583289061616622868, −2.36001044194066787842847866437, −2.18822706532143536372194136385, −2.01160056477925509586882304847, −1.95468742437509288471998477070, −1.94632938036208238474901335433, −1.68459243807640908031823664641, −1.56241475623698960277861460661, −1.47936506738700072968361347365, −1.40215187499432110833649530849, −1.33582640264442453639243839816, −0.49268780319222486721864611312, −0.45496733228175796857600698588, −0.19305191914037496694585225321,
0.19305191914037496694585225321, 0.45496733228175796857600698588, 0.49268780319222486721864611312, 1.33582640264442453639243839816, 1.40215187499432110833649530849, 1.47936506738700072968361347365, 1.56241475623698960277861460661, 1.68459243807640908031823664641, 1.94632938036208238474901335433, 1.95468742437509288471998477070, 2.01160056477925509586882304847, 2.18822706532143536372194136385, 2.36001044194066787842847866437, 2.48876129939583289061616622868, 3.00079794300709524015225920262, 3.15664164261279725242054342114, 3.16695376294254708959347273991, 3.27359972307916229728672016501, 3.31793641467848323867108434559, 3.49834041296306632400281689801, 3.53224979246201419615844734503, 3.61227013324088992179442689442, 3.67253730929176898173946710656, 4.09594833455184229815712036545, 4.18976783455498990837800475709
Plot not available for L-functions of degree greater than 10.
