| L(s) = 1 | + 2-s + 3-s + 6-s − 5·7-s − 8-s + 4·11-s + 8·13-s − 5·14-s − 16-s − 3·17-s − 6·19-s − 5·21-s + 4·22-s − 7·23-s − 24-s + 8·26-s − 27-s + 8·29-s + 5·31-s + 4·33-s − 3·34-s − 2·37-s − 6·38-s + 8·39-s + 14·41-s − 5·42-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1.20·11-s + 2.21·13-s − 1.33·14-s − 1/4·16-s − 0.727·17-s − 1.37·19-s − 1.09·21-s + 0.852·22-s − 1.45·23-s − 0.204·24-s + 1.56·26-s − 0.192·27-s + 1.48·29-s + 0.898·31-s + 0.696·33-s − 0.514·34-s − 0.328·37-s − 0.973·38-s + 1.28·39-s + 2.18·41-s − 0.771·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.757836453\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.757836453\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 12 T + 83 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−10.13107181354361460055953823594, −9.559759764051861110127683069233, −9.287047571880463630538903988012, −8.780002676284681995349139662872, −8.617530226995158245551434302029, −8.319576605369241117188942051794, −7.56514754076904702347420389497, −7.02389647558746370875847266360, −6.41163176683544734035480011699, −6.32614360972854453090207017458, −5.99676937004566357556620420972, −5.75936402153922773320879643468, −4.47614740692558688018804823087, −4.36246107132561924897415241811, −3.79764884993598755077774260435, −3.63469731031097482624793299935, −2.85872547013559820690040815149, −2.55034009698614313194599238537, −1.60570822393362871205643876305, −0.66015070190360516515504171734,
0.66015070190360516515504171734, 1.60570822393362871205643876305, 2.55034009698614313194599238537, 2.85872547013559820690040815149, 3.63469731031097482624793299935, 3.79764884993598755077774260435, 4.36246107132561924897415241811, 4.47614740692558688018804823087, 5.75936402153922773320879643468, 5.99676937004566357556620420972, 6.32614360972854453090207017458, 6.41163176683544734035480011699, 7.02389647558746370875847266360, 7.56514754076904702347420389497, 8.319576605369241117188942051794, 8.617530226995158245551434302029, 8.780002676284681995349139662872, 9.287047571880463630538903988012, 9.559759764051861110127683069233, 10.13107181354361460055953823594
