| L(s) = 1 | − 2-s + 4·5-s + 7-s + 8-s − 4·10-s − 4·11-s − 7·13-s − 14-s − 16-s + 7·17-s − 2·19-s + 4·22-s − 23-s + 2·25-s + 7·26-s − 2·29-s − 18·31-s − 7·34-s + 4·35-s + 2·37-s + 2·38-s + 4·40-s + 2·41-s + 5·43-s + 46-s − 12·47-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·5-s + 0.377·7-s + 0.353·8-s − 1.26·10-s − 1.20·11-s − 1.94·13-s − 0.267·14-s − 1/4·16-s + 1.69·17-s − 0.458·19-s + 0.852·22-s − 0.208·23-s + 2/5·25-s + 1.37·26-s − 0.371·29-s − 3.23·31-s − 1.20·34-s + 0.676·35-s + 0.328·37-s + 0.324·38-s + 0.632·40-s + 0.312·41-s + 0.762·43-s + 0.147·46-s − 1.75·47-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.092268493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092268493\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.669533033871346904008043128175, −9.460471315512146920822304815418, −9.006570598722941376458067428915, −8.276296823344769309324498057909, −7.966622955765090332313248848926, −7.64279738716173650968807363233, −7.45397594130239479069616350947, −6.73305917898861959478450693490, −6.50666402101929150346338032102, −5.57367479694585242909144318197, −5.56227794120664417953217923564, −5.17096030392605402313517005075, −5.05077766017886537264560511044, −3.98652754648966586795570880068, −3.73934886432153024356773800070, −2.76527997354497507826947511820, −2.46011244694000534717974466018, −1.86448015206884076550505144257, −1.63513405545175008033898498993, −0.42698813041559905090219435230,
0.42698813041559905090219435230, 1.63513405545175008033898498993, 1.86448015206884076550505144257, 2.46011244694000534717974466018, 2.76527997354497507826947511820, 3.73934886432153024356773800070, 3.98652754648966586795570880068, 5.05077766017886537264560511044, 5.17096030392605402313517005075, 5.56227794120664417953217923564, 5.57367479694585242909144318197, 6.50666402101929150346338032102, 6.73305917898861959478450693490, 7.45397594130239479069616350947, 7.64279738716173650968807363233, 7.966622955765090332313248848926, 8.276296823344769309324498057909, 9.006570598722941376458067428915, 9.460471315512146920822304815418, 9.669533033871346904008043128175
