| L(s) = 1 | + 2·3-s + 2·5-s + 6·7-s + 3·9-s + 2·11-s + 2·13-s + 4·15-s − 8·17-s + 10·19-s + 12·21-s − 2·23-s + 3·25-s + 4·27-s − 4·29-s + 6·31-s + 4·33-s + 12·35-s + 10·37-s + 4·39-s − 4·41-s + 16·43-s + 6·45-s + 2·47-s + 13·49-s − 16·51-s − 8·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s + 2.26·7-s + 9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s − 1.94·17-s + 2.29·19-s + 2.61·21-s − 0.417·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s + 1.07·31-s + 0.696·33-s + 2.02·35-s + 1.64·37-s + 0.640·39-s − 0.624·41-s + 2.43·43-s + 0.894·45-s + 0.291·47-s + 13/7·49-s − 2.24·51-s − 1.09·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.66088298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.66088298\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 59 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 10 T + 51 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 83 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 119 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 95 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 131 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 143 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 72 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 307 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 202 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.129347280251093242153330823246, −8.019663219313853471754741685138, −7.69816142998773634572117553536, −7.57057733184152510942458826939, −6.94156256406943812965732465440, −6.58297550159701544326427455596, −6.17058105511240596548364711953, −5.92628983817436065660355130575, −5.19666943101418800582618283153, −5.09536655742963787837109325148, −4.62726337760161498988752723831, −4.40771652018598276140934925013, −3.72328784874369764541000643198, −3.70387052193895692338936175229, −2.81065045947876252454656091501, −2.48825745595824878734006289163, −2.15800128589694955721755914532, −1.67900826087782248659776801165, −1.20455765263658779787805711143, −0.922397691644222319169880871688,
0.922397691644222319169880871688, 1.20455765263658779787805711143, 1.67900826087782248659776801165, 2.15800128589694955721755914532, 2.48825745595824878734006289163, 2.81065045947876252454656091501, 3.70387052193895692338936175229, 3.72328784874369764541000643198, 4.40771652018598276140934925013, 4.62726337760161498988752723831, 5.09536655742963787837109325148, 5.19666943101418800582618283153, 5.92628983817436065660355130575, 6.17058105511240596548364711953, 6.58297550159701544326427455596, 6.94156256406943812965732465440, 7.57057733184152510942458826939, 7.69816142998773634572117553536, 8.019663219313853471754741685138, 8.129347280251093242153330823246
