| L(s) = 1 | + 2·5-s − 7-s − 11-s − 2·13-s − 3·17-s + 2·19-s − 7·23-s + 3·25-s + 4·29-s − 2·31-s − 2·35-s + 5·37-s − 3·41-s + 8·43-s − 14·47-s − 5·49-s + 13·53-s − 2·55-s − 16·59-s − 61-s − 4·65-s + 8·67-s − 17·71-s − 8·73-s + 77-s + 5·79-s − 24·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.301·11-s − 0.554·13-s − 0.727·17-s + 0.458·19-s − 1.45·23-s + 3/5·25-s + 0.742·29-s − 0.359·31-s − 0.338·35-s + 0.821·37-s − 0.468·41-s + 1.21·43-s − 2.04·47-s − 5/7·49-s + 1.78·53-s − 0.269·55-s − 2.08·59-s − 0.128·61-s − 0.496·65-s + 0.977·67-s − 2.01·71-s − 0.936·73-s + 0.113·77-s + 0.562·79-s − 2.63·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 50 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 110 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 140 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 17 T + 206 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T - 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 280 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 216 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−7.37126576520460063186449832926, −7.36260235897820516648837672457, −6.74215509750075020064421085369, −6.44517083362422060305877615240, −6.26127953776175934227163672828, −5.77847447493658038854936035321, −5.57738046483224934607791576224, −5.21494013140140729895441808495, −4.65719440074040608045652462579, −4.50725610219306882771222678169, −4.08485578316610031022038986518, −3.65851103780854802152982847288, −2.97480035732674762098551703764, −2.91023564875863976948880606652, −2.40541538142864417246006335655, −2.02770007626584602906840732852, −1.46890181188502866701761087545, −1.14333726250537489383586625927, 0, 0,
1.14333726250537489383586625927, 1.46890181188502866701761087545, 2.02770007626584602906840732852, 2.40541538142864417246006335655, 2.91023564875863976948880606652, 2.97480035732674762098551703764, 3.65851103780854802152982847288, 4.08485578316610031022038986518, 4.50725610219306882771222678169, 4.65719440074040608045652462579, 5.21494013140140729895441808495, 5.57738046483224934607791576224, 5.77847447493658038854936035321, 6.26127953776175934227163672828, 6.44517083362422060305877615240, 6.74215509750075020064421085369, 7.36260235897820516648837672457, 7.37126576520460063186449832926
