| L(s) = 1 | − 8·5-s + 36·17-s − 2·25-s + 8·29-s − 144·37-s + 36·41-s + 50·49-s − 88·53-s + 144·61-s + 164·73-s − 288·85-s + 252·89-s + 220·97-s − 184·101-s + 288·109-s + 252·113-s + 194·121-s + 344·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 8/5·5-s + 2.11·17-s − 0.0799·25-s + 8/29·29-s − 3.89·37-s + 0.878·41-s + 1.02·49-s − 1.66·53-s + 2.36·61-s + 2.24·73-s − 3.38·85-s + 2.83·89-s + 2.26·97-s − 1.82·101-s + 2.64·109-s + 2.23·113-s + 1.60·121-s + 2.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.441·145-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.259787883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259787883\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 670 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 430 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 72 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 190 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2690 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3074 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8546 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 8354 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8594 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3550 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
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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
−8.792965041384278656761581316620, −8.585081239161908475465186722612, −8.172006174319927943381951482063, −7.889804166674299664863384780534, −7.48932624125498120374026764466, −7.22341721462457816482821629702, −6.89984128577460705456718224200, −6.29711854971928586587776725532, −5.81421425970055171167312193691, −5.49218143516174428892555809273, −4.84401590323109193613491700823, −4.84198425325790838178367550805, −3.86398730551703527437524691908, −3.76872076468839524172496825342, −3.35989044661583345359048628213, −3.08941070924581785416980646561, −2.03365005775707414458225334903, −1.82023954881505262170754939332, −0.67716615386652887863828849783, −0.59342339736474747198062134827,
0.59342339736474747198062134827, 0.67716615386652887863828849783, 1.82023954881505262170754939332, 2.03365005775707414458225334903, 3.08941070924581785416980646561, 3.35989044661583345359048628213, 3.76872076468839524172496825342, 3.86398730551703527437524691908, 4.84198425325790838178367550805, 4.84401590323109193613491700823, 5.49218143516174428892555809273, 5.81421425970055171167312193691, 6.29711854971928586587776725532, 6.89984128577460705456718224200, 7.22341721462457816482821629702, 7.48932624125498120374026764466, 7.889804166674299664863384780534, 8.172006174319927943381951482063, 8.585081239161908475465186722612, 8.792965041384278656761581316620
