| L(s) = 1 | + 24·41-s + 64·79-s − 14·81-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 3.74·41-s + 7.20·79-s − 1.55·81-s − 6.90·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.060939179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.060939179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 + 7 T^{4} + p^{4} T^{8} )^{2} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 19 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2}( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | \( ( 1 + 383 T^{4} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 7 T + p T^{2} )^{4}( 1 + 7 T + p T^{2} )^{4} \) |
| 23 | \( ( 1 - 1054 T^{4} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2638 T^{4} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 3 T + p T^{2} )^{8} \) |
| 43 | \( ( 1 - 3214 T^{4} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 2302 T^{4} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 4174 T^{4} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 - 1753 T^{4} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 5617 T^{4} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 8 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 - 10297 T^{4} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.92841584662510935693893195843, −3.88397635245664513026673544791, −3.67032049662817605886231406423, −3.59681055063111323737490577453, −3.55431582292447091007903329469, −3.41758483200198836041067915410, −3.25209832377657234200961714255, −3.08207303931617567067596077429, −2.93014766240247356971742117124, −2.83040396112453970421681622264, −2.47823277575108148503985928954, −2.43560720639912955013014501765, −2.42091462544983558098771119960, −2.29930437406557449526828395140, −2.25310032782426223090508583253, −2.20209083742000614580826404743, −1.72869128764872927887494032261, −1.51870360235355995842671640212, −1.42512862514712417690793785871, −1.23791528484242469205629514386, −1.23282357790255448567440887827, −0.830471342245449491155021444035, −0.76769493844489884768752366698, −0.34672066626031302673102280118, −0.32829662459581199653572173332,
0.32829662459581199653572173332, 0.34672066626031302673102280118, 0.76769493844489884768752366698, 0.830471342245449491155021444035, 1.23282357790255448567440887827, 1.23791528484242469205629514386, 1.42512862514712417690793785871, 1.51870360235355995842671640212, 1.72869128764872927887494032261, 2.20209083742000614580826404743, 2.25310032782426223090508583253, 2.29930437406557449526828395140, 2.42091462544983558098771119960, 2.43560720639912955013014501765, 2.47823277575108148503985928954, 2.83040396112453970421681622264, 2.93014766240247356971742117124, 3.08207303931617567067596077429, 3.25209832377657234200961714255, 3.41758483200198836041067915410, 3.55431582292447091007903329469, 3.59681055063111323737490577453, 3.67032049662817605886231406423, 3.88397635245664513026673544791, 3.92841584662510935693893195843
Plot not available for L-functions of degree greater than 10.
