| L(s) = 1 | − 3-s − 5-s + 3·9-s − 3·11-s − 2·13-s + 15-s + 3·17-s − 2·19-s + 6·23-s − 8·27-s − 18·29-s − 8·31-s + 3·33-s + 10·37-s + 2·39-s + 4·43-s − 3·45-s + 3·47-s − 3·51-s + 3·55-s + 2·57-s − 12·59-s − 8·61-s + 2·65-s − 8·67-s − 6·69-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 9-s − 0.904·11-s − 0.554·13-s + 0.258·15-s + 0.727·17-s − 0.458·19-s + 1.25·23-s − 1.53·27-s − 3.34·29-s − 1.43·31-s + 0.522·33-s + 1.64·37-s + 0.320·39-s + 0.609·43-s − 0.447·45-s + 0.437·47-s − 0.420·51-s + 0.404·55-s + 0.264·57-s − 1.56·59-s − 1.02·61-s + 0.248·65-s − 0.977·67-s − 0.722·69-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7744914156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7744914156\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−10.19878233630553201070847709145, −9.695375697406569904243552067044, −9.584924796427693982190559879827, −8.889497265194187587586100394818, −8.777127298541232457126279446556, −7.80463771312474312150857378387, −7.50389391736109657239854899061, −7.28627167712099794396599284422, −7.27998785116000433031061430270, −6.18359118117925877405878665415, −5.72011459006091455358417945390, −5.68524353764833429902033358503, −4.98870038823146990883262032440, −4.32970782677682468646644018440, −4.23875086691898894471816145757, −3.34791081269071475089753149529, −3.04149919230460280942316065031, −2.02138164466295000112059816738, −1.63147896362715371417168117469, −0.41746101808479425691552841503,
0.41746101808479425691552841503, 1.63147896362715371417168117469, 2.02138164466295000112059816738, 3.04149919230460280942316065031, 3.34791081269071475089753149529, 4.23875086691898894471816145757, 4.32970782677682468646644018440, 4.98870038823146990883262032440, 5.68524353764833429902033358503, 5.72011459006091455358417945390, 6.18359118117925877405878665415, 7.27998785116000433031061430270, 7.28627167712099794396599284422, 7.50389391736109657239854899061, 7.80463771312474312150857378387, 8.777127298541232457126279446556, 8.889497265194187587586100394818, 9.584924796427693982190559879827, 9.695375697406569904243552067044, 10.19878233630553201070847709145
