| L(s) = 1 | + 2·3-s − 4-s − 3·9-s − 2·12-s + 16-s + 6·17-s + 25-s − 14·27-s + 12·29-s + 3·36-s + 2·43-s + 2·48-s + 13·49-s + 12·51-s + 16·61-s − 64-s − 6·68-s + 2·75-s + 16·79-s − 4·81-s + 24·87-s − 100-s + 24·101-s + 8·103-s + 24·107-s + 14·108-s − 12·113-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s − 9-s − 0.577·12-s + 1/4·16-s + 1.45·17-s + 1/5·25-s − 2.69·27-s + 2.22·29-s + 1/2·36-s + 0.304·43-s + 0.288·48-s + 13/7·49-s + 1.68·51-s + 2.04·61-s − 1/8·64-s − 0.727·68-s + 0.230·75-s + 1.80·79-s − 4/9·81-s + 2.57·87-s − 0.0999·100-s + 2.38·101-s + 0.788·103-s + 2.32·107-s + 1.34·108-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.862435550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.862435550\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.90001938655246642161124448681, −11.47962023623496683249556426007, −10.68951228124473413516396354642, −10.25682981267986509509626086362, −9.970806440026238400885315201692, −9.161260634330154200706631741996, −9.026687183871298662204841434953, −8.524661109040951760923924459211, −8.085411398851038431044936968833, −7.77756593576225514711222864746, −7.19424624485425059378327960563, −6.33863028403466947556436283928, −5.96957496143121267328805667713, −5.21954265905149702204053665503, −4.94658345547980238986834259477, −3.80091000690223610783049956115, −3.61364056707757015726153296503, −2.72840214598476615671868185953, −2.42549519787529325700206170165, −0.955561772345536587695390546105,
0.955561772345536587695390546105, 2.42549519787529325700206170165, 2.72840214598476615671868185953, 3.61364056707757015726153296503, 3.80091000690223610783049956115, 4.94658345547980238986834259477, 5.21954265905149702204053665503, 5.96957496143121267328805667713, 6.33863028403466947556436283928, 7.19424624485425059378327960563, 7.77756593576225514711222864746, 8.085411398851038431044936968833, 8.524661109040951760923924459211, 9.026687183871298662204841434953, 9.161260634330154200706631741996, 9.970806440026238400885315201692, 10.25682981267986509509626086362, 10.68951228124473413516396354642, 11.47962023623496683249556426007, 11.90001938655246642161124448681
