| L(s) = 1 | − 3·2-s + 4·4-s + 3·5-s − 3·8-s − 9·10-s + 3·11-s + 3·13-s + 3·16-s − 6·17-s + 12·20-s − 9·22-s + 9·23-s + 5·25-s − 9·26-s + 9·29-s − 6·31-s − 6·32-s + 18·34-s + 14·37-s − 9·40-s + 3·41-s − 43-s + 12·44-s − 27·46-s − 15·50-s + 12·52-s + 9·55-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 1.34·5-s − 1.06·8-s − 2.84·10-s + 0.904·11-s + 0.832·13-s + 3/4·16-s − 1.45·17-s + 2.68·20-s − 1.91·22-s + 1.87·23-s + 25-s − 1.76·26-s + 1.67·29-s − 1.07·31-s − 1.06·32-s + 3.08·34-s + 2.30·37-s − 1.42·40-s + 0.468·41-s − 0.152·43-s + 1.80·44-s − 3.98·46-s − 2.12·50-s + 1.66·52-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106388903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106388903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.697778134602527532931990691281, −9.452343792465186527144318567497, −9.027047913070435534133398151140, −8.715630154397928085371836593257, −8.359132464722967223231435292206, −8.306552609302574962413431944206, −7.31308050027186436270258386943, −7.13965319030429974490050212771, −6.64056105573029073589598917437, −6.37804935225052383484084021853, −5.71219349024419316037135419274, −5.60072337566334986200317027605, −4.59573602895056667858747484345, −4.47294253263176121643558856221, −3.60755652207958569346407357934, −2.94488852275526794249830348716, −2.39512660561262032965554588682, −1.81921505335063145846772891630, −1.07148710939418932926165939560, −0.798811610972492510804057422554,
0.798811610972492510804057422554, 1.07148710939418932926165939560, 1.81921505335063145846772891630, 2.39512660561262032965554588682, 2.94488852275526794249830348716, 3.60755652207958569346407357934, 4.47294253263176121643558856221, 4.59573602895056667858747484345, 5.60072337566334986200317027605, 5.71219349024419316037135419274, 6.37804935225052383484084021853, 6.64056105573029073589598917437, 7.13965319030429974490050212771, 7.31308050027186436270258386943, 8.306552609302574962413431944206, 8.359132464722967223231435292206, 8.715630154397928085371836593257, 9.027047913070435534133398151140, 9.452343792465186527144318567497, 9.697778134602527532931990691281
