| L(s) = 1 | + 4·5-s − 9-s − 8·11-s − 4·19-s + 11·25-s − 12·29-s − 12·31-s − 4·45-s − 49-s − 32·55-s + 8·59-s − 4·61-s + 16·71-s + 32·79-s + 81-s − 32·89-s − 16·95-s + 8·99-s + 16·101-s − 36·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 48·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s − 2.41·11-s − 0.917·19-s + 11/5·25-s − 2.22·29-s − 2.15·31-s − 0.596·45-s − 1/7·49-s − 4.31·55-s + 1.04·59-s − 0.512·61-s + 1.89·71-s + 3.60·79-s + 1/9·81-s − 3.39·89-s − 1.64·95-s + 0.804·99-s + 1.59·101-s − 3.44·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.98·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.403052917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403052917\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−9.473085835099832135961826760589, −9.309383403710638666001786121605, −8.892317637351689119864639239854, −8.238221615209891668067861805353, −8.129618539359861725128170546132, −7.46076405834372518269636748419, −7.31389556620405462878996554263, −6.59041345150838864876647244373, −6.35471301963850703238367892455, −5.68340562035333934168063884294, −5.48754715163448808142478910701, −5.16279447382091566237105160711, −5.00808821292139965791964853524, −3.96093171659562900994942581441, −3.70456521129806073514636610767, −2.74431134969039036229484213345, −2.64614280749033305500312946816, −1.84652139555756762753248856884, −1.84559586549780823792609125113, −0.41002701862452815638068003388,
0.41002701862452815638068003388, 1.84559586549780823792609125113, 1.84652139555756762753248856884, 2.64614280749033305500312946816, 2.74431134969039036229484213345, 3.70456521129806073514636610767, 3.96093171659562900994942581441, 5.00808821292139965791964853524, 5.16279447382091566237105160711, 5.48754715163448808142478910701, 5.68340562035333934168063884294, 6.35471301963850703238367892455, 6.59041345150838864876647244373, 7.31389556620405462878996554263, 7.46076405834372518269636748419, 8.129618539359861725128170546132, 8.238221615209891668067861805353, 8.892317637351689119864639239854, 9.309383403710638666001786121605, 9.473085835099832135961826760589
