| L(s) = 1 | + 4·5-s + 10·9-s + 4·13-s + 10·25-s + 40·45-s − 2·49-s + 32·61-s + 16·65-s + 57·81-s + 24·101-s − 48·113-s + 40·117-s + 10·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 42·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 10/3·9-s + 1.10·13-s + 2·25-s + 5.96·45-s − 2/7·49-s + 4.09·61-s + 1.98·65-s + 19/3·81-s + 2.38·101-s − 4.51·113-s + 3.69·117-s + 0.909·121-s + 1.78·125-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 − 3.23·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.75967891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.75967891\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 31 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 157 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.49051681812150881339043703274, −6.20838970236099942508768902929, −6.07793787880874468408217800109, −5.84956584016079561769464848289, −5.65489848596982778514886741454, −5.16489170313982877004621355497, −5.13266520978742060342733363730, −5.01913356546128779552949983040, −4.94234415056642729475000337388, −4.40079957013832487509431369616, −4.27109815892030525015288315851, −4.05177984686490142623216957577, −3.88791233560929109298455440758, −3.65221426277436102736282216059, −3.50877788849952376035444208309, −3.11909605249643632339684740063, −2.64757529371516388409776438228, −2.59378659542787943891821898824, −2.14200426539424490131240110686, −1.98160142511764907854629566829, −1.81906892529811825722170615204, −1.31857953092311406613232116498, −1.18744683661151100937232727541, −1.09033472637065608661244287888, −0.53844007915995601454196239166,
0.53844007915995601454196239166, 1.09033472637065608661244287888, 1.18744683661151100937232727541, 1.31857953092311406613232116498, 1.81906892529811825722170615204, 1.98160142511764907854629566829, 2.14200426539424490131240110686, 2.59378659542787943891821898824, 2.64757529371516388409776438228, 3.11909605249643632339684740063, 3.50877788849952376035444208309, 3.65221426277436102736282216059, 3.88791233560929109298455440758, 4.05177984686490142623216957577, 4.27109815892030525015288315851, 4.40079957013832487509431369616, 4.94234415056642729475000337388, 5.01913356546128779552949983040, 5.13266520978742060342733363730, 5.16489170313982877004621355497, 5.65489848596982778514886741454, 5.84956584016079561769464848289, 6.07793787880874468408217800109, 6.20838970236099942508768902929, 6.49051681812150881339043703274
