| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s − 2·7-s − 4·8-s + 3·9-s + 4·11-s − 6·12-s − 2·13-s + 4·14-s + 5·16-s − 2·17-s − 6·18-s − 2·19-s + 4·21-s − 8·22-s − 2·23-s + 8·24-s + 4·26-s − 4·27-s − 6·28-s + 2·31-s − 6·32-s − 8·33-s + 4·34-s + 9·36-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s + 1.20·11-s − 1.73·12-s − 0.554·13-s + 1.06·14-s + 5/4·16-s − 0.485·17-s − 1.41·18-s − 0.458·19-s + 0.872·21-s − 1.70·22-s − 0.417·23-s + 1.63·24-s + 0.784·26-s − 0.769·27-s − 1.13·28-s + 0.359·31-s − 1.06·32-s − 1.39·33-s + 0.685·34-s + 3/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 51 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 100 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 135 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 164 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 135 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 167 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 56 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.589637006218528387421201326961, −8.360220202259788747413491715663, −7.70982633546656140732920970351, −7.59551879178427233119901295986, −6.86463295501935493794246110714, −6.76204246059534951640029166572, −6.50337156095344798948573195491, −6.12746734437435725474151843130, −5.69493695338968508555443379806, −5.28864338973732717151319425623, −4.60046546227079517422315404741, −4.41123081895737540513744296132, −3.58457526565963067757022467920, −3.50897135866188363673771118030, −2.51491091965069771385375490574, −2.31132162016088562257512473367, −1.36647173771941590272376626945, −1.22005894131790457823118301979, 0, 0,
1.22005894131790457823118301979, 1.36647173771941590272376626945, 2.31132162016088562257512473367, 2.51491091965069771385375490574, 3.50897135866188363673771118030, 3.58457526565963067757022467920, 4.41123081895737540513744296132, 4.60046546227079517422315404741, 5.28864338973732717151319425623, 5.69493695338968508555443379806, 6.12746734437435725474151843130, 6.50337156095344798948573195491, 6.76204246059534951640029166572, 6.86463295501935493794246110714, 7.59551879178427233119901295986, 7.70982633546656140732920970351, 8.360220202259788747413491715663, 8.589637006218528387421201326961
