| L(s) = 1 | + 3·5-s − 5·7-s + 9·11-s + 6·13-s + 12·17-s + 9·23-s + 5·25-s − 18·29-s + 9·31-s − 15·35-s + 2·37-s − 3·41-s − 10·43-s − 6·47-s + 18·49-s + 27·55-s + 6·59-s + 24·61-s + 18·65-s − 2·67-s − 45·77-s − 14·79-s + 6·83-s + 36·85-s − 18·89-s − 30·91-s + 12·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.88·7-s + 2.71·11-s + 1.66·13-s + 2.91·17-s + 1.87·23-s + 25-s − 3.34·29-s + 1.61·31-s − 2.53·35-s + 0.328·37-s − 0.468·41-s − 1.52·43-s − 0.875·47-s + 18/7·49-s + 3.64·55-s + 0.781·59-s + 3.07·61-s + 2.23·65-s − 0.244·67-s − 5.12·77-s − 1.57·79-s + 0.658·83-s + 3.90·85-s − 1.90·89-s − 3.14·91-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.627386375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.627386375\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 18 T + 137 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 253 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 134 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.507470013495837683527479355557, −8.971169305644872959816237085069, −8.448331693819944350159991563158, −8.416836886736875482224823691529, −7.44155411449298065825338825650, −7.15985897898664530680364618036, −6.70316113740652510031768526515, −6.49831050372623773386126757140, −6.14803186350318474106581110272, −5.69327221825384313571380470349, −5.54717702203720572825408905269, −5.06113459125429457780400616757, −4.07749960917803374156680789587, −3.75832181604404260756758000886, −3.35588428468028598045394777770, −3.33266659538549056966651776267, −2.53168421958939540825964796948, −1.59385207033353707983056220150, −1.28318635902816524425711520816, −0.885021576774499407782085664412,
0.885021576774499407782085664412, 1.28318635902816524425711520816, 1.59385207033353707983056220150, 2.53168421958939540825964796948, 3.33266659538549056966651776267, 3.35588428468028598045394777770, 3.75832181604404260756758000886, 4.07749960917803374156680789587, 5.06113459125429457780400616757, 5.54717702203720572825408905269, 5.69327221825384313571380470349, 6.14803186350318474106581110272, 6.49831050372623773386126757140, 6.70316113740652510031768526515, 7.15985897898664530680364618036, 7.44155411449298065825338825650, 8.416836886736875482224823691529, 8.448331693819944350159991563158, 8.971169305644872959816237085069, 9.507470013495837683527479355557
