| L(s) = 1 | − 2-s + 2·3-s − 2·4-s − 2·6-s + 3·8-s + 3·9-s − 4·12-s + 8·13-s + 16-s + 3·17-s − 3·18-s + 5·19-s − 7·23-s + 6·24-s − 5·25-s − 8·26-s + 4·27-s + 15·29-s + 31-s − 2·32-s − 3·34-s − 6·36-s − 7·37-s − 5·38-s + 16·39-s − 5·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s − 4-s − 0.816·6-s + 1.06·8-s + 9-s − 1.15·12-s + 2.21·13-s + 1/4·16-s + 0.727·17-s − 0.707·18-s + 1.14·19-s − 1.45·23-s + 1.22·24-s − 25-s − 1.56·26-s + 0.769·27-s + 2.78·29-s + 0.179·31-s − 0.353·32-s − 0.514·34-s − 36-s − 1.15·37-s − 0.811·38-s + 2.56·39-s − 0.780·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75220929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75220929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.734795016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.734795016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 13 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 7 T + 57 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 15 T + 113 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 - T - 39 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 75 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 57 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 15 T + 139 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 13 T + 153 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 105 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 141 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - T + 165 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 29 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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
−8.135018439475707136148569988074, −7.75080754378283671500298971590, −7.55899069559779410453407256388, −6.86788874770543397037375358453, −6.76163448756393245173886025389, −6.29732025880020580153408031847, −5.81292915114095099830026623942, −5.60781780777939226368934020894, −5.03584804140819360065737925962, −4.87677970282889573947362294086, −4.14079228678678275849108231307, −3.97193097280322542207863412540, −3.65656278811474393312339990066, −3.50850146456227558243930107994, −2.72844873640226872150572682207, −2.57843577214141261246596660078, −1.80445012584167116594210500994, −1.42445820823652862455748715383, −0.892951721473480871712300133031, −0.62804332673305916072648043846,
0.62804332673305916072648043846, 0.892951721473480871712300133031, 1.42445820823652862455748715383, 1.80445012584167116594210500994, 2.57843577214141261246596660078, 2.72844873640226872150572682207, 3.50850146456227558243930107994, 3.65656278811474393312339990066, 3.97193097280322542207863412540, 4.14079228678678275849108231307, 4.87677970282889573947362294086, 5.03584804140819360065737925962, 5.60781780777939226368934020894, 5.81292915114095099830026623942, 6.29732025880020580153408031847, 6.76163448756393245173886025389, 6.86788874770543397037375358453, 7.55899069559779410453407256388, 7.75080754378283671500298971590, 8.135018439475707136148569988074
