| L(s) = 1 | − 2·5-s + 7-s + 3·11-s + 2·13-s − 9·17-s − 8·19-s − 9·23-s + 3·25-s − 6·29-s − 2·31-s − 2·35-s + 37-s − 3·41-s + 10·43-s − 5·49-s − 9·53-s − 6·55-s − 6·59-s + 19·61-s − 4·65-s − 8·67-s + 3·71-s − 8·73-s + 3·77-s − 5·79-s − 18·83-s + 18·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s + 0.904·11-s + 0.554·13-s − 2.18·17-s − 1.83·19-s − 1.87·23-s + 3/5·25-s − 1.11·29-s − 0.359·31-s − 0.338·35-s + 0.164·37-s − 0.468·41-s + 1.52·43-s − 5/7·49-s − 1.23·53-s − 0.809·55-s − 0.781·59-s + 2.43·61-s − 0.496·65-s − 0.977·67-s + 0.356·71-s − 0.936·73-s + 0.341·77-s − 0.562·79-s − 1.97·83-s + 1.95·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 76 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 118 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 19 T + 204 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 136 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T - 42 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 18 T + 214 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 9 T + 124 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 29 T + 396 T^{2} + 29 p T^{3} + p^{2} T^{4} \) |
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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
−8.601657371804789759220859752892, −8.582041332167730929035646662541, −7.942869290403312098260705554020, −7.905310436139609331133831169540, −7.19845619347951974394342910805, −6.75261411107191203432796943815, −6.54305940218421449164801088586, −6.26488849128986717078509648254, −5.56954528696220806206304385373, −5.38642467498859461681626274934, −4.41350125105966121611828288026, −4.28745281808839688389508315082, −4.01366997485287848565259445128, −3.83657146738409054673809423623, −2.81315838585871549979452267273, −2.49946825346163621314334396795, −1.70880617446467909880194200828, −1.51862037406253076562418287376, 0, 0,
1.51862037406253076562418287376, 1.70880617446467909880194200828, 2.49946825346163621314334396795, 2.81315838585871549979452267273, 3.83657146738409054673809423623, 4.01366997485287848565259445128, 4.28745281808839688389508315082, 4.41350125105966121611828288026, 5.38642467498859461681626274934, 5.56954528696220806206304385373, 6.26488849128986717078509648254, 6.54305940218421449164801088586, 6.75261411107191203432796943815, 7.19845619347951974394342910805, 7.905310436139609331133831169540, 7.942869290403312098260705554020, 8.582041332167730929035646662541, 8.601657371804789759220859752892
