| L(s) = 1 | − 4-s + 5-s − 11-s + 16-s − 5·19-s − 20-s − 4·25-s − 6·29-s − 31-s + 9·41-s + 44-s + 5·49-s − 55-s + 14·61-s − 64-s − 71-s + 5·76-s − 15·79-s + 80-s − 9·81-s + 15·89-s − 5·95-s + 4·100-s + 19·101-s + 20·109-s + 6·116-s − 15·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.447·5-s − 0.301·11-s + 1/4·16-s − 1.14·19-s − 0.223·20-s − 4/5·25-s − 1.11·29-s − 0.179·31-s + 1.40·41-s + 0.150·44-s + 5/7·49-s − 0.134·55-s + 1.79·61-s − 1/8·64-s − 0.118·71-s + 0.573·76-s − 1.68·79-s + 0.111·80-s − 81-s + 1.58·89-s − 0.512·95-s + 2/5·100-s + 1.89·101-s + 1.91·109-s + 0.557·116-s − 1.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6834152878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6834152878\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 150 T^{2} + p^{2} T^{4} \) |
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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
−12.97300761202007026711745745915, −12.48015169428370421085238309500, −11.58970666047779625110098362727, −11.09080844592056808048366738307, −10.31061285899280809288523356227, −9.878830696598072344068565692663, −9.107759636504157791839335091252, −8.627910123400402193922707762082, −7.80232759810327212643626620231, −7.14372402130757995244587466183, −6.10113281784815210264840570215, −5.59836080241083055223088887588, −4.57544865124138979668811884308, −3.72787597549157883263338949238, −2.25054967550044909031450405591,
2.25054967550044909031450405591, 3.72787597549157883263338949238, 4.57544865124138979668811884308, 5.59836080241083055223088887588, 6.10113281784815210264840570215, 7.14372402130757995244587466183, 7.80232759810327212643626620231, 8.627910123400402193922707762082, 9.107759636504157791839335091252, 9.878830696598072344068565692663, 10.31061285899280809288523356227, 11.09080844592056808048366738307, 11.58970666047779625110098362727, 12.48015169428370421085238309500, 12.97300761202007026711745745915
