| L(s) = 1 | + 2·2-s + 2·4-s + 8·11-s + 13-s − 4·16-s + 16·22-s − 5·23-s − 25-s + 2·26-s − 8·32-s + 3·37-s + 16·44-s − 10·46-s − 3·47-s − 4·49-s − 2·50-s + 2·52-s + 12·59-s − 8·61-s − 8·64-s + 20·71-s − 15·73-s + 6·74-s + 8·83-s − 10·92-s − 6·94-s − 29·97-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 2.41·11-s + 0.277·13-s − 16-s + 3.41·22-s − 1.04·23-s − 1/5·25-s + 0.392·26-s − 1.41·32-s + 0.493·37-s + 2.41·44-s − 1.47·46-s − 0.437·47-s − 4/7·49-s − 0.282·50-s + 0.277·52-s + 1.56·59-s − 1.02·61-s − 64-s + 2.37·71-s − 1.75·73-s + 0.697·74-s + 0.878·83-s − 1.04·92-s − 0.618·94-s − 2.94·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.997687906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.997687906\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 39 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−9.969243933921348492547657204926, −9.627394795563388305054935257136, −9.084088215862560007926166425656, −8.623902929902130405952538525125, −8.014555157714648410204883089796, −7.18896434827089969280326885561, −6.67991335175063798929937795442, −6.22944499550437841682075757107, −5.89020410590446340826539463631, −5.06865421877077047201623851854, −4.42192165394394478703558002363, −3.81306540456660432418783632101, −3.59943963050718989444876063849, −2.49396410196780533934073260913, −1.47411948399879935963617089019,
1.47411948399879935963617089019, 2.49396410196780533934073260913, 3.59943963050718989444876063849, 3.81306540456660432418783632101, 4.42192165394394478703558002363, 5.06865421877077047201623851854, 5.89020410590446340826539463631, 6.22944499550437841682075757107, 6.67991335175063798929937795442, 7.18896434827089969280326885561, 8.014555157714648410204883089796, 8.623902929902130405952538525125, 9.084088215862560007926166425656, 9.627394795563388305054935257136, 9.969243933921348492547657204926
