| L(s) = 1 | + 16·11-s − 16-s − 64·71-s − 18·81-s + 116·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s − 16·176-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 4.82·11-s − 1/4·16-s − 7.59·71-s − 2·81-s + 10.5·121-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 − 1.20·176-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955850059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955850059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 734 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 - 1294 T^{4} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 334 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
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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
−9.295788130223123997749919651171, −8.949074337796219816279989960370, −8.731718288912379459667430769406, −8.711639087669041416502681045798, −8.658223868755804939964032895364, −7.83180944624360715170230145558, −7.70305955566681878430446794179, −7.27470388019706962283762256508, −7.06461819598525994554228135893, −6.66142962221300964512146884740, −6.61283229222771205671144169486, −6.43782600054211081230415846159, −5.79199757591857807238924552386, −5.72789353503646374271817312350, −5.63387458556468777830125829925, −4.56349747045914290790628961683, −4.48817535752594670137064032761, −4.26820059794563016061310760884, −4.12133940717376354354410348413, −3.50040511643722665526440664288, −3.24913354249812247416201433180, −2.87310546007828154061144350583, −1.94423318089632559465389065916, −1.43264616941654813267669169944, −1.28937223116315859726322458995,
1.28937223116315859726322458995, 1.43264616941654813267669169944, 1.94423318089632559465389065916, 2.87310546007828154061144350583, 3.24913354249812247416201433180, 3.50040511643722665526440664288, 4.12133940717376354354410348413, 4.26820059794563016061310760884, 4.48817535752594670137064032761, 4.56349747045914290790628961683, 5.63387458556468777830125829925, 5.72789353503646374271817312350, 5.79199757591857807238924552386, 6.43782600054211081230415846159, 6.61283229222771205671144169486, 6.66142962221300964512146884740, 7.06461819598525994554228135893, 7.27470388019706962283762256508, 7.70305955566681878430446794179, 7.83180944624360715170230145558, 8.658223868755804939964032895364, 8.711639087669041416502681045798, 8.731718288912379459667430769406, 8.949074337796219816279989960370, 9.295788130223123997749919651171
