| L(s) = 1 | − 3·3-s + 4-s + 2·7-s + 2·9-s − 11-s − 3·12-s + 13-s − 3·16-s − 17-s − 13·19-s − 6·21-s + 6·27-s + 2·28-s − 3·29-s − 4·31-s + 3·33-s + 2·36-s − 3·39-s + 7·41-s − 12·43-s − 44-s + 9·47-s + 9·48-s − 6·49-s + 3·51-s + 52-s + 12·53-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1/2·4-s + 0.755·7-s + 2/3·9-s − 0.301·11-s − 0.866·12-s + 0.277·13-s − 3/4·16-s − 0.242·17-s − 2.98·19-s − 1.30·21-s + 1.15·27-s + 0.377·28-s − 0.557·29-s − 0.718·31-s + 0.522·33-s + 1/3·36-s − 0.480·39-s + 1.09·41-s − 1.82·43-s − 0.150·44-s + 1.31·47-s + 1.29·48-s − 6/7·49-s + 0.420·51-s + 0.138·52-s + 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | | \( 1 \) |
| 241 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 15 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 13 T + 79 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + p T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 93 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 113 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 74 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 105 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 47 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 187 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 13 T + 177 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 214 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.990826264581264827518262314735, −7.26910898099974260737272675032, −7.16806921806607797838383517961, −6.60626158602514357383050825672, −6.49416157457007506918104019709, −6.19341451408584374871711381940, −5.70514619004060113225561410790, −5.49598776227145022730825417455, −5.09035517761714538665271601876, −4.77723661938273495426333475556, −4.34780276917949215552720148122, −3.89609400861748581854196560595, −3.74166410150022803688847032439, −2.79343411246067938360707577060, −2.29595540104545699431139219537, −2.22941467887679852569963123646, −1.57525249054104748864286375215, −0.906722131916748905367403451711, 0, 0,
0.906722131916748905367403451711, 1.57525249054104748864286375215, 2.22941467887679852569963123646, 2.29595540104545699431139219537, 2.79343411246067938360707577060, 3.74166410150022803688847032439, 3.89609400861748581854196560595, 4.34780276917949215552720148122, 4.77723661938273495426333475556, 5.09035517761714538665271601876, 5.49598776227145022730825417455, 5.70514619004060113225561410790, 6.19341451408584374871711381940, 6.49416157457007506918104019709, 6.60626158602514357383050825672, 7.16806921806607797838383517961, 7.26910898099974260737272675032, 7.990826264581264827518262314735
