| L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s − 12·11-s + 35·16-s + 48·22-s + 8·23-s − 16·25-s − 4·29-s − 56·32-s − 8·37-s − 8·43-s − 120·44-s − 32·46-s + 64·50-s − 44·53-s + 16·58-s + 84·64-s + 32·67-s − 16·71-s + 32·74-s + 9·81-s + 32·86-s + 240·88-s + 80·92-s − 160·100-s + 176·106-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s − 3.61·11-s + 35/4·16-s + 10.2·22-s + 1.66·23-s − 3.19·25-s − 0.742·29-s − 9.89·32-s − 1.31·37-s − 1.21·43-s − 18.0·44-s − 4.71·46-s + 9.05·50-s − 6.04·53-s + 2.10·58-s + 21/2·64-s + 3.90·67-s − 1.89·71-s + 3.71·74-s + 81-s + 3.45·86-s + 25.5·88-s + 8.34·92-s − 16·100-s + 17.0·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - p^{2} T^{4} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 111 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 16 T^{2} + 159 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 48 T^{2} + 1106 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 1647 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 5094 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 4 T + 87 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 6447 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 22 T + 215 T^{2} + 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 124 T^{2} + 7734 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 6054 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 + 84 T^{2} + 9350 T^{4} + 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 76 T^{2} + 2934 T^{4} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 52 T^{2} + 15318 T^{4} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 180 T^{2} + 23846 T^{4} + 180 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.83290587034305677385892126942, −6.60644204235076614481918381811, −6.27251021573548855525687818809, −6.11405823134162359480203476711, −5.99509346719385495076084568756, −5.57913066045414892990766163708, −5.41241779859835878049553594903, −5.31804811176987880050623746339, −5.16071609587773319687725882551, −5.03070247285147016270879787367, −4.68939974954955118796342984812, −4.26093328116140821596426534090, −4.25197077755369779391870467396, −3.59445183369259526530429553371, −3.46730870109208702697315065487, −3.30994597794425797223401039226, −3.21728074640835944222391820699, −2.63505792135426607401679036234, −2.62171885007143553708471718568, −2.40920318405761553814801951664, −2.18808283983612251085083565671, −1.82884122888780038748348227221, −1.61238091855406002567305279284, −1.20730331634176497959639156771, −1.16211424586762208055039876015, 0, 0, 0, 0,
1.16211424586762208055039876015, 1.20730331634176497959639156771, 1.61238091855406002567305279284, 1.82884122888780038748348227221, 2.18808283983612251085083565671, 2.40920318405761553814801951664, 2.62171885007143553708471718568, 2.63505792135426607401679036234, 3.21728074640835944222391820699, 3.30994597794425797223401039226, 3.46730870109208702697315065487, 3.59445183369259526530429553371, 4.25197077755369779391870467396, 4.26093328116140821596426534090, 4.68939974954955118796342984812, 5.03070247285147016270879787367, 5.16071609587773319687725882551, 5.31804811176987880050623746339, 5.41241779859835878049553594903, 5.57913066045414892990766163708, 5.99509346719385495076084568756, 6.11405823134162359480203476711, 6.27251021573548855525687818809, 6.60644204235076614481918381811, 6.83290587034305677385892126942
