| L(s) = 1 | − 6·3-s + 22·5-s − 268·7-s + 37·9-s − 242·11-s − 1.23e3·13-s − 132·15-s − 124·17-s + 1.94e3·19-s + 1.60e3·21-s − 3.34e3·23-s − 3.90e3·25-s − 1.68e3·27-s + 6.57e3·29-s − 2.49e3·31-s + 1.45e3·33-s − 5.89e3·35-s + 1.46e4·37-s + 7.39e3·39-s − 6.50e3·41-s + 1.16e4·43-s + 814·45-s − 3.68e4·47-s + 2.47e4·49-s + 744·51-s + 3.29e3·53-s − 5.32e3·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.393·5-s − 2.06·7-s + 0.152·9-s − 0.603·11-s − 2.02·13-s − 0.151·15-s − 0.104·17-s + 1.23·19-s + 0.795·21-s − 1.31·23-s − 1.24·25-s − 0.445·27-s + 1.45·29-s − 0.466·31-s + 0.232·33-s − 0.813·35-s + 1.76·37-s + 0.778·39-s − 0.604·41-s + 0.959·43-s + 0.0599·45-s − 2.43·47-s + 1.47·49-s + 0.0400·51-s + 0.160·53-s − 0.237·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495616 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4899117007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4899117007\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 2 p T - T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 22 T + 4387 T^{2} - 22 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 268 T + 47106 T^{2} + 268 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 1232 T + 1109642 T^{2} + 1232 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 1747894 T^{2} + 124 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1944 T + 5420326 T^{2} - 1944 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3346 T + 13945039 T^{2} + 3346 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6576 T + 48815578 T^{2} - 6576 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2498 T - 9451633 T^{2} + 2498 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14674 T + 188136827 T^{2} - 14674 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6504 T + 129433522 T^{2} + 6504 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 11628 T + 122110426 T^{2} - 11628 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 36816 T + 662392414 T^{2} + 36816 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3292 T + 183140302 T^{2} - 3292 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12126 T + 1296864967 T^{2} - 12126 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 73128 T + 2869460074 T^{2} + 73128 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 29334 T + 1425739759 T^{2} - 29334 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 46122 T + 3691685527 T^{2} - 46122 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8240 T + 3846470690 T^{2} - 8240 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14780 T + 5848676514 T^{2} - 14780 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 74564 T + 8948999626 T^{2} + 74564 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 33698 T + 11242447699 T^{2} + 33698 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 126162 T + 16112874451 T^{2} - 126162 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.738979556084376896834493882915, −9.720630562123374379559436087140, −9.408304952579798658649605854244, −8.681381960347730654940120687157, −7.977907351200612859533277499045, −7.59498329118134936485904339990, −7.39301827240293199703959053750, −6.67946315678667231719971175892, −6.25659110578381914397103172377, −6.08481685117465706562125065651, −5.44650951679884388796027416252, −4.99905357547837363451960990712, −4.48529607378281089115996878437, −3.88225025640208308197991571819, −3.05720567650389293819942072751, −3.01859498039630301095370983237, −2.26750516379119393300338834812, −1.76665569747907293649432819174, −0.67460626137236580394911963060, −0.21695042599729016595842671508,
0.21695042599729016595842671508, 0.67460626137236580394911963060, 1.76665569747907293649432819174, 2.26750516379119393300338834812, 3.01859498039630301095370983237, 3.05720567650389293819942072751, 3.88225025640208308197991571819, 4.48529607378281089115996878437, 4.99905357547837363451960990712, 5.44650951679884388796027416252, 6.08481685117465706562125065651, 6.25659110578381914397103172377, 6.67946315678667231719971175892, 7.39301827240293199703959053750, 7.59498329118134936485904339990, 7.977907351200612859533277499045, 8.681381960347730654940120687157, 9.408304952579798658649605854244, 9.720630562123374379559436087140, 9.738979556084376896834493882915
