| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·8-s + 3·9-s − 4·11-s + 6·12-s − 6·13-s + 5·16-s − 2·17-s − 6·18-s + 8·19-s + 8·22-s − 6·23-s − 8·24-s + 12·26-s + 4·27-s − 10·29-s + 6·31-s − 6·32-s − 8·33-s + 4·34-s + 9·36-s + 8·37-s − 16·38-s − 12·39-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.41·8-s + 9-s − 1.20·11-s + 1.73·12-s − 1.66·13-s + 5/4·16-s − 0.485·17-s − 1.41·18-s + 1.83·19-s + 1.70·22-s − 1.25·23-s − 1.63·24-s + 2.35·26-s + 0.769·27-s − 1.85·29-s + 1.07·31-s − 1.06·32-s − 1.39·33-s + 0.685·34-s + 3/2·36-s + 1.31·37-s − 2.59·38-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54022500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 51 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 109 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 125 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 99 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 144 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 144 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 213 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.74602347659869281482990310371, −7.66920806050289540110001828022, −7.29705850022234309087358260224, −7.04179204885918823875940091589, −6.42572435023661323931638582910, −6.16610090117366087918978054723, −5.52516475090098826695516691950, −5.51890611097641053897623794760, −4.82156744546584382620253167052, −4.55898128570346078399449768362, −3.97183861648782052606208895118, −3.69431172530884496409180206380, −2.92812909381330448273411689339, −2.77593223191264811479013311001, −2.33966041802478290014560773669, −2.31897474883982250228949028555, −1.29401060343727134137564740177, −1.26282817164240804467242802351, 0, 0,
1.26282817164240804467242802351, 1.29401060343727134137564740177, 2.31897474883982250228949028555, 2.33966041802478290014560773669, 2.77593223191264811479013311001, 2.92812909381330448273411689339, 3.69431172530884496409180206380, 3.97183861648782052606208895118, 4.55898128570346078399449768362, 4.82156744546584382620253167052, 5.51890611097641053897623794760, 5.52516475090098826695516691950, 6.16610090117366087918978054723, 6.42572435023661323931638582910, 7.04179204885918823875940091589, 7.29705850022234309087358260224, 7.66920806050289540110001828022, 7.74602347659869281482990310371
