| L(s) = 1 | − 2·3-s + 2·5-s − 6·11-s − 2·13-s − 4·15-s − 10·19-s + 6·23-s + 3·25-s + 2·27-s + 4·29-s + 6·31-s + 12·33-s − 8·37-s + 4·39-s − 16·41-s − 2·43-s + 16·47-s − 2·49-s − 16·53-s − 12·55-s + 20·57-s − 14·59-s + 4·61-s − 4·65-s + 4·67-s − 12·69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 1.80·11-s − 0.554·13-s − 1.03·15-s − 2.29·19-s + 1.25·23-s + 3/5·25-s + 0.384·27-s + 0.742·29-s + 1.07·31-s + 2.08·33-s − 1.31·37-s + 0.640·39-s − 2.49·41-s − 0.304·43-s + 2.33·47-s − 2/7·49-s − 2.19·53-s − 1.61·55-s + 2.64·57-s − 1.82·59-s + 0.512·61-s − 0.496·65-s + 0.488·67-s − 1.44·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17305600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17305600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 60 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 68 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 14 T + 140 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T - 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 218 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 134 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.153695986224601568686807444565, −7.924556127440171673570594437290, −7.51016938370051918669702147862, −6.92297408956779928419352381173, −6.56278735134248137142832891559, −6.52962708155734549656583518105, −5.81384029597582723347815597172, −5.80890462462597810055224435736, −5.18324571691467376107017036004, −4.96873051243735728186729570101, −4.64062699937574084895990812403, −4.40809897639979275248218922249, −3.29076342628133821139963155438, −3.28537677782255646499112458168, −2.55562416575041674176406491642, −2.23523813590377150952993827436, −1.79459983712075140583831359747, −1.00258231606128310147078879795, 0, 0,
1.00258231606128310147078879795, 1.79459983712075140583831359747, 2.23523813590377150952993827436, 2.55562416575041674176406491642, 3.28537677782255646499112458168, 3.29076342628133821139963155438, 4.40809897639979275248218922249, 4.64062699937574084895990812403, 4.96873051243735728186729570101, 5.18324571691467376107017036004, 5.80890462462597810055224435736, 5.81384029597582723347815597172, 6.52962708155734549656583518105, 6.56278735134248137142832891559, 6.92297408956779928419352381173, 7.51016938370051918669702147862, 7.924556127440171673570594437290, 8.153695986224601568686807444565
