| L(s) = 1 | − 2·5-s − 2·7-s − 2·11-s + 6·13-s − 2·17-s − 4·19-s + 3·25-s − 4·29-s + 4·31-s + 4·35-s + 4·41-s − 2·43-s − 4·47-s + 2·49-s + 12·53-s + 4·55-s + 8·61-s − 12·65-s − 24·71-s + 2·73-s + 4·77-s − 28·79-s − 26·83-s + 4·85-s − 4·89-s − 12·91-s + 8·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s − 0.603·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 3/5·25-s − 0.742·29-s + 0.718·31-s + 0.676·35-s + 0.624·41-s − 0.304·43-s − 0.583·47-s + 2/7·49-s + 1.64·53-s + 0.539·55-s + 1.02·61-s − 1.48·65-s − 2.84·71-s + 0.234·73-s + 0.455·77-s − 3.15·79-s − 2.85·83-s + 0.433·85-s − 0.423·89-s − 1.25·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 134 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 26 T + 322 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−7.49256583691718421742061804681, −7.39316010566265295861166509601, −6.94844655272811034755422056027, −6.75982877247418833198041529366, −6.23425062675689203714115654617, −5.97567911107094094442145142256, −5.52218478960576133596550647706, −5.51858187238615866264427522285, −4.54659573847595067111368138853, −4.49551645685559091874652789170, −4.02517410178739606943380095615, −3.86910579489095356437486703493, −3.23267967074121657541667616645, −3.00118151932031925998973631019, −2.59179614535535003165811060940, −2.05509134447070223145644445014, −1.38804338963901062696261015755, −1.02680469340857162915502034470, 0, 0,
1.02680469340857162915502034470, 1.38804338963901062696261015755, 2.05509134447070223145644445014, 2.59179614535535003165811060940, 3.00118151932031925998973631019, 3.23267967074121657541667616645, 3.86910579489095356437486703493, 4.02517410178739606943380095615, 4.49551645685559091874652789170, 4.54659573847595067111368138853, 5.51858187238615866264427522285, 5.52218478960576133596550647706, 5.97567911107094094442145142256, 6.23425062675689203714115654617, 6.75982877247418833198041529366, 6.94844655272811034755422056027, 7.39316010566265295861166509601, 7.49256583691718421742061804681
