| L(s) = 1 | + 2·2-s + 9·4-s + 30·8-s + 2·9-s − 4·11-s + 60·16-s + 4·18-s − 8·22-s − 52·23-s + 5·25-s − 88·29-s + 198·32-s + 18·36-s − 28·37-s − 136·43-s − 36·44-s − 104·46-s + 10·50-s + 68·53-s − 176·58-s + 411·64-s − 28·67-s + 248·71-s + 60·72-s − 56·74-s − 76·79-s + 81·81-s + ⋯ |
| L(s) = 1 | + 2-s + 9/4·4-s + 15/4·8-s + 2/9·9-s − 0.363·11-s + 15/4·16-s + 2/9·18-s − 0.363·22-s − 2.26·23-s + 1/5·25-s − 3.03·29-s + 6.18·32-s + 1/2·36-s − 0.756·37-s − 3.16·43-s − 0.818·44-s − 2.26·46-s + 1/5·50-s + 1.28·53-s − 3.03·58-s + 6.42·64-s − 0.417·67-s + 3.49·71-s + 5/6·72-s − 0.756·74-s − 0.962·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.607820942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.607820942\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T - 3 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 7 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )( 1 + 4 T + 7 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 117 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 142 T^{2} - 63357 T^{4} - 142 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 542 T^{2} + 163443 T^{4} + 542 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T + 147 T^{2} + 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 958 T^{2} - 5757 T^{4} - 958 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 14 T - 1173 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2642 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 34 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 + 3698 T^{2} + 8795523 T^{4} + 3698 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 34 T - 1653 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 5342 T^{2} + 16419603 T^{4} + 5342 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 1378 T^{2} - 11946957 T^{4} - 1378 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T - 4293 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 7778 T^{2} + 32099043 T^{4} + 7778 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 12158 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 15122 T^{2} + 165932643 T^{4} + 15122 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 18098 T^{2} + p^{4} T^{4} )^{2} \) |
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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
−8.363991216542749219789582265175, −8.360224229059625110274042644431, −8.017315210354615845561494198186, −7.61022066592897011094966974948, −7.38424908468801504469808526276, −7.19161228971948224195511734877, −7.02388554908605775252201190503, −6.84319940149168898929416707854, −6.42536202998211152525665797295, −6.11332230067157001671873791664, −5.87020734211045585980949158990, −5.53194114099407474107481844356, −5.43194497350807717004152102539, −4.72114926844044746608003665839, −4.71329397836631365307018809704, −4.52293093362063188319181736114, −3.88997045158379891500890460614, −3.57832550164358794703555500922, −3.50496026266258321000482213658, −3.04912542249195318936393808313, −2.11366412189244002473895940568, −2.11318072116315370897137878087, −2.00037996981255593958463649111, −1.56030048769393479063871158816, −0.56155143235569557449100441323,
0.56155143235569557449100441323, 1.56030048769393479063871158816, 2.00037996981255593958463649111, 2.11318072116315370897137878087, 2.11366412189244002473895940568, 3.04912542249195318936393808313, 3.50496026266258321000482213658, 3.57832550164358794703555500922, 3.88997045158379891500890460614, 4.52293093362063188319181736114, 4.71329397836631365307018809704, 4.72114926844044746608003665839, 5.43194497350807717004152102539, 5.53194114099407474107481844356, 5.87020734211045585980949158990, 6.11332230067157001671873791664, 6.42536202998211152525665797295, 6.84319940149168898929416707854, 7.02388554908605775252201190503, 7.19161228971948224195511734877, 7.38424908468801504469808526276, 7.61022066592897011094966974948, 8.017315210354615845561494198186, 8.360224229059625110274042644431, 8.363991216542749219789582265175
