| L(s) = 1 | − 3·2-s + 4·4-s + 4·7-s − 3·8-s − 9-s + 2·11-s + 4·13-s − 12·14-s + 3·16-s + 3·17-s + 3·18-s − 6·22-s − 2·23-s − 12·26-s + 16·28-s + 12·29-s + 3·31-s − 6·32-s − 9·34-s − 4·36-s − 15·37-s − 15·41-s − 43-s + 8·44-s + 6·46-s + 13·47-s + 3·49-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s + 1.51·7-s − 1.06·8-s − 1/3·9-s + 0.603·11-s + 1.10·13-s − 3.20·14-s + 3/4·16-s + 0.727·17-s + 0.707·18-s − 1.27·22-s − 0.417·23-s − 2.35·26-s + 3.02·28-s + 2.22·29-s + 0.538·31-s − 1.06·32-s − 1.54·34-s − 2/3·36-s − 2.46·37-s − 2.34·41-s − 0.152·43-s + 1.20·44-s + 0.884·46-s + 1.89·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9025305275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9025305275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 4 T + 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 53 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 15 T + 119 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 137 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 135 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 161 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 41 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 93 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 18 T + 203 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 9 T + 155 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 142 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17 T + 265 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.063478435842347692585199520105, −7.77857227967649788125774006460, −7.39883610481573311883408009279, −6.97215607957067174994238822349, −6.84779370315140410773908081322, −6.24280760099609814104711435923, −6.00809376938263900948328488445, −5.60396282023249303941941109653, −5.03671459334692084818376305645, −4.98597118099573634361415085573, −4.48110911184354852977182520115, −3.85304799684076774528147974675, −3.64232794652694099581733285167, −3.22032625915699265015250281192, −2.60110369019768769603385459769, −2.18811997409870235101322460703, −1.53129035439211855026790440799, −1.33293718011767349634791484893, −1.05182148982571412818263169108, −0.37098842880183033455771216198,
0.37098842880183033455771216198, 1.05182148982571412818263169108, 1.33293718011767349634791484893, 1.53129035439211855026790440799, 2.18811997409870235101322460703, 2.60110369019768769603385459769, 3.22032625915699265015250281192, 3.64232794652694099581733285167, 3.85304799684076774528147974675, 4.48110911184354852977182520115, 4.98597118099573634361415085573, 5.03671459334692084818376305645, 5.60396282023249303941941109653, 6.00809376938263900948328488445, 6.24280760099609814104711435923, 6.84779370315140410773908081322, 6.97215607957067174994238822349, 7.39883610481573311883408009279, 7.77857227967649788125774006460, 8.063478435842347692585199520105
