L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 6·9-s − 4·11-s + 5·16-s − 12·18-s − 8·22-s − 12·23-s − 8·25-s − 2·29-s + 6·32-s − 18·36-s − 8·37-s − 8·43-s − 12·44-s − 24·46-s − 16·50-s − 12·53-s − 4·58-s + 7·64-s + 8·67-s − 4·71-s − 24·72-s − 16·74-s − 24·79-s + 27·81-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2·9-s − 1.20·11-s + 5/4·16-s − 2.82·18-s − 1.70·22-s − 2.50·23-s − 8/5·25-s − 0.371·29-s + 1.06·32-s − 3·36-s − 1.31·37-s − 1.21·43-s − 1.80·44-s − 3.53·46-s − 2.26·50-s − 1.64·53-s − 0.525·58-s + 7/8·64-s + 0.977·67-s − 0.474·71-s − 2.82·72-s − 1.85·74-s − 2.70·79-s + 3·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8076964 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 60 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 176 T^{2} + p^{2} T^{4} \) |
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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
−8.391380315410183321566307985131, −8.134192960944930627699696299254, −7.75125707506951203302501292996, −7.69206300215806924993208924290, −6.77477756733577158503456457859, −6.68087583947254292067490665931, −5.98782726668821337417103714650, −5.85590396186397951401981424018, −5.44380380667879276791351701699, −5.34721497262056777570880583713, −4.69835287911109903932785863065, −4.26471953308040053037934590732, −3.71068413147365303744513046607, −3.48091181158175284480969721410, −2.84903477660818846975483656186, −2.62397009510143994678621541863, −1.94548932682159096552640825297, −1.70828781096723838430056037580, 0, 0,
1.70828781096723838430056037580, 1.94548932682159096552640825297, 2.62397009510143994678621541863, 2.84903477660818846975483656186, 3.48091181158175284480969721410, 3.71068413147365303744513046607, 4.26471953308040053037934590732, 4.69835287911109903932785863065, 5.34721497262056777570880583713, 5.44380380667879276791351701699, 5.85590396186397951401981424018, 5.98782726668821337417103714650, 6.68087583947254292067490665931, 6.77477756733577158503456457859, 7.69206300215806924993208924290, 7.75125707506951203302501292996, 8.134192960944930627699696299254, 8.391380315410183321566307985131