| L(s) = 1 | − 2·4-s + 2·5-s + 2·13-s + 3·16-s + 2·17-s − 4·20-s + 3·25-s − 2·37-s + 2·41-s − 49-s − 4·52-s − 2·53-s − 4·64-s + 4·65-s − 4·68-s − 2·73-s + 6·80-s + 4·85-s − 6·100-s − 2·113-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯ |
| L(s) = 1 | − 2·4-s + 2·5-s + 2·13-s + 3·16-s + 2·17-s − 4·20-s + 3·25-s − 2·37-s + 2·41-s − 49-s − 4·52-s − 2·53-s − 4·64-s + 4·65-s − 4·68-s − 2·73-s + 6·80-s + 4·85-s − 6·100-s − 2·113-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 4·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.147391591\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147391591\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−7.37741019449462285087211132918, −6.84194261714711779480708818302, −6.75738431060650757701392096164, −6.47785703853219884531475605215, −6.35593963701832303572856363312, −5.91665754843749741734688080901, −5.79741031208453825217921125763, −5.73902641479683288670924875470, −5.41133926647758289575073481390, −5.35936631764707582864410491532, −5.16653767258138405860674440609, −4.69670895471886942507610139756, −4.42373353536815510266667469571, −4.35329586426174275798526098259, −4.21543338030380175554953567885, −3.48996876030461619486339069932, −3.46644045804636655364032790428, −3.21849441578673391053630187454, −3.20521755991001567859526013285, −2.75331563927132718503450088724, −2.12756519643561008156849201862, −1.95515965875115808831887195138, −1.29610807322623552332265508911, −1.27229463794011922222899848491, −1.06980856282376242341092973028,
1.06980856282376242341092973028, 1.27229463794011922222899848491, 1.29610807322623552332265508911, 1.95515965875115808831887195138, 2.12756519643561008156849201862, 2.75331563927132718503450088724, 3.20521755991001567859526013285, 3.21849441578673391053630187454, 3.46644045804636655364032790428, 3.48996876030461619486339069932, 4.21543338030380175554953567885, 4.35329586426174275798526098259, 4.42373353536815510266667469571, 4.69670895471886942507610139756, 5.16653767258138405860674440609, 5.35936631764707582864410491532, 5.41133926647758289575073481390, 5.73902641479683288670924875470, 5.79741031208453825217921125763, 5.91665754843749741734688080901, 6.35593963701832303572856363312, 6.47785703853219884531475605215, 6.75738431060650757701392096164, 6.84194261714711779480708818302, 7.37741019449462285087211132918
