| L(s) = 1 | − 2·2-s + 3·4-s + 4·5-s − 2·7-s − 4·8-s − 8·10-s + 2·11-s − 6·13-s + 4·14-s + 5·16-s + 14·17-s − 2·19-s + 12·20-s − 4·22-s + 2·23-s + 5·25-s + 12·26-s − 6·28-s + 10·29-s + 6·31-s − 6·32-s − 28·34-s − 8·35-s + 4·37-s + 4·38-s − 16·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.78·5-s − 0.755·7-s − 1.41·8-s − 2.52·10-s + 0.603·11-s − 1.66·13-s + 1.06·14-s + 5/4·16-s + 3.39·17-s − 0.458·19-s + 2.68·20-s − 0.852·22-s + 0.417·23-s + 25-s + 2.35·26-s − 1.13·28-s + 1.85·29-s + 1.07·31-s − 1.06·32-s − 4.80·34-s − 1.35·35-s + 0.657·37-s + 0.648·38-s − 2.52·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.842703433\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.842703433\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 44 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 92 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 10 T + 156 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 20 T + 230 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 20 T + 243 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 20 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 76 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 139 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.764491109229933810634573918241, −9.695210987152636214352896835533, −9.340457984359283769773850794833, −9.214678306585608311542763007920, −8.289468455539870886670604642057, −8.073173611060736142642857294085, −7.50150875808826303243150444821, −7.39123445470273608523883788881, −6.52320000086489884650777301024, −6.42458775102284963770257394605, −5.86013973546035986635768945154, −5.70335391592220612498138019709, −4.98664153712767081207825785621, −4.56348919810310256435831712296, −3.48244777814721618789283460379, −3.13081215238899425447998646359, −2.37442263818242509803958140776, −2.30127166572978617752789484775, −1.09808247313066073318935895647, −0.953241572069657719294986291341,
0.953241572069657719294986291341, 1.09808247313066073318935895647, 2.30127166572978617752789484775, 2.37442263818242509803958140776, 3.13081215238899425447998646359, 3.48244777814721618789283460379, 4.56348919810310256435831712296, 4.98664153712767081207825785621, 5.70335391592220612498138019709, 5.86013973546035986635768945154, 6.42458775102284963770257394605, 6.52320000086489884650777301024, 7.39123445470273608523883788881, 7.50150875808826303243150444821, 8.073173611060736142642857294085, 8.289468455539870886670604642057, 9.214678306585608311542763007920, 9.340457984359283769773850794833, 9.695210987152636214352896835533, 9.764491109229933810634573918241
