| L(s) = 1 | − 2-s − 3·3-s − 2·4-s + 3·6-s − 2·7-s + 3·8-s + 2·9-s − 11-s + 6·12-s + 2·13-s + 2·14-s + 16-s − 17-s − 2·18-s − 2·19-s + 6·21-s + 22-s + 2·23-s − 9·24-s − 2·26-s + 6·27-s + 4·28-s + 5·29-s − 31-s − 2·32-s + 3·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 4-s + 1.22·6-s − 0.755·7-s + 1.06·8-s + 2/3·9-s − 0.301·11-s + 1.73·12-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.458·19-s + 1.30·21-s + 0.213·22-s + 0.417·23-s − 1.83·24-s − 0.392·26-s + 1.15·27-s + 0.755·28-s + 0.928·29-s − 0.179·31-s − 0.353·32-s + 0.522·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11055625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11055625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + T - 39 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14 T + 103 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 213 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 51 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 103 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 101 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 97 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 197 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 10 T + 198 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 183 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.475569185301752602272888963220, −8.188818028822848562646414430951, −7.78423118090213607678709270571, −7.39441645472446870862481231843, −6.66600759056467009514544291918, −6.41973048140314218067175341169, −6.41121063004378722085474049561, −5.70316929645525785283293333630, −5.46884003464350543941550648534, −5.15402494522891045906164095787, −4.50991255011411563092255451281, −4.43888352429981856865681249455, −3.88506001445648598244046437493, −3.32203349441969414693836035246, −2.69951712371192310590330129090, −2.36299871874618798418398276476, −1.10172245036876523690399916982, −1.03172919208215681637551132015, 0, 0,
1.03172919208215681637551132015, 1.10172245036876523690399916982, 2.36299871874618798418398276476, 2.69951712371192310590330129090, 3.32203349441969414693836035246, 3.88506001445648598244046437493, 4.43888352429981856865681249455, 4.50991255011411563092255451281, 5.15402494522891045906164095787, 5.46884003464350543941550648534, 5.70316929645525785283293333630, 6.41121063004378722085474049561, 6.41973048140314218067175341169, 6.66600759056467009514544291918, 7.39441645472446870862481231843, 7.78423118090213607678709270571, 8.188818028822848562646414430951, 8.475569185301752602272888963220
