| L(s) = 1 | + 2·3-s + 2·4-s + 3·5-s + 3·9-s + 4·12-s + 2·13-s + 6·15-s + 6·17-s + 7·19-s + 6·20-s − 3·23-s + 5·25-s + 10·27-s − 18·29-s − 5·31-s + 6·36-s − 2·37-s + 4·39-s − 12·41-s − 2·43-s + 9·45-s − 3·47-s + 12·51-s + 4·52-s + 9·53-s + 14·57-s + 12·60-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 4-s + 1.34·5-s + 9-s + 1.15·12-s + 0.554·13-s + 1.54·15-s + 1.45·17-s + 1.60·19-s + 1.34·20-s − 0.625·23-s + 25-s + 1.92·27-s − 3.34·29-s − 0.898·31-s + 36-s − 0.328·37-s + 0.640·39-s − 1.87·41-s − 0.304·43-s + 1.34·45-s − 0.437·47-s + 1.68·51-s + 0.554·52-s + 1.23·53-s + 1.85·57-s + 1.54·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.210779717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.210779717\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.60320733949262905995285434806, −10.20178575872346732636842172104, −9.943476455264955985501598289319, −9.547652114237072681773877091954, −9.099520615524474934016293733985, −8.666662431911456873433076355555, −8.307352463725448282309612326081, −7.40020227169092827934891774212, −7.32382781730063247368061341758, −7.15422656240831436384898795626, −6.06474884243324355243380018483, −6.05956903679745416831384629871, −5.34970223126175432920954939460, −5.05728078844996327807484628377, −4.00330740096828896103718494942, −3.33655919731409294451986272758, −3.20996845857271698451291459010, −2.35575860272932616176503063076, −1.72524943218544239780109093382, −1.45239219827050149743360769574,
1.45239219827050149743360769574, 1.72524943218544239780109093382, 2.35575860272932616176503063076, 3.20996845857271698451291459010, 3.33655919731409294451986272758, 4.00330740096828896103718494942, 5.05728078844996327807484628377, 5.34970223126175432920954939460, 6.05956903679745416831384629871, 6.06474884243324355243380018483, 7.15422656240831436384898795626, 7.32382781730063247368061341758, 7.40020227169092827934891774212, 8.307352463725448282309612326081, 8.666662431911456873433076355555, 9.099520615524474934016293733985, 9.547652114237072681773877091954, 9.943476455264955985501598289319, 10.20178575872346732636842172104, 10.60320733949262905995285434806
