| L(s) = 1 | − 128·4-s + 286·7-s + 1.22e4·16-s + 3.12e4·25-s − 3.66e4·28-s + 1.78e5·37-s − 2.22e5·43-s − 3.58e4·49-s − 1.04e6·64-s + 3.45e5·67-s − 4.09e5·79-s − 4.00e6·100-s + 4.34e6·109-s + 3.51e6·112-s − 3.54e6·121-s + 127-s + 131-s + 137-s + 139-s − 2.28e7·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9.39e6·169-s + 2.85e7·172-s + ⋯ |
| L(s) = 1 | − 2·4-s + 0.833·7-s + 3·16-s + 2·25-s − 1.66·28-s + 3.52·37-s − 2.80·43-s − 0.304·49-s − 4·64-s + 1.14·67-s − 0.830·79-s − 4·100-s + 3.35·109-s + 2.50·112-s − 2·121-s − 7.04·148-s − 1.94·169-s + 5.60·172-s + 1.66·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.476510458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476510458\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 286 T + p^{6} T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 506 T + p^{6} T^{2} )( 1 + 506 T + p^{6} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 10582 T + p^{6} T^{2} )( 1 + 10582 T + p^{6} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 35282 T + p^{6} T^{2} )( 1 + 35282 T + p^{6} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 89206 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 111386 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 420838 T + p^{6} T^{2} )( 1 + 420838 T + p^{6} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 172874 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{6} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 638066 T + p^{6} T^{2} )( 1 + 638066 T + p^{6} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 204622 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 56446 T + p^{6} T^{2} )( 1 + 56446 T + p^{6} T^{2} ) \) |
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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
−14.14199223025322877616778101703, −13.32045698637609446972387338312, −12.97771421229965521450819681308, −12.68664176319695695390675865768, −11.71117893917093761801511377892, −11.26446990328814786963790138097, −10.38787846317477670059584092767, −9.928130378637771342270114305799, −9.275429945673257532050972362973, −8.781369914662678164013051483620, −8.130055251316737024001435228300, −7.84515636991541616962108028828, −6.70164854838705590275019359667, −5.78961228504370970289519241539, −4.86311610881280886619194008741, −4.75897615703778227721602044130, −3.85255999471248975855030853558, −2.90103904038935372051226921364, −1.33290097498636670857109930530, −0.58431789581049588792543677085,
0.58431789581049588792543677085, 1.33290097498636670857109930530, 2.90103904038935372051226921364, 3.85255999471248975855030853558, 4.75897615703778227721602044130, 4.86311610881280886619194008741, 5.78961228504370970289519241539, 6.70164854838705590275019359667, 7.84515636991541616962108028828, 8.130055251316737024001435228300, 8.781369914662678164013051483620, 9.275429945673257532050972362973, 9.928130378637771342270114305799, 10.38787846317477670059584092767, 11.26446990328814786963790138097, 11.71117893917093761801511377892, 12.68664176319695695390675865768, 12.97771421229965521450819681308, 13.32045698637609446972387338312, 14.14199223025322877616778101703
