| L(s) = 1 | − 2·5-s + 7-s − 6·13-s − 2·17-s + 4·19-s − 4·23-s − 25-s − 10·29-s + 8·31-s − 2·35-s + 6·37-s − 2·41-s − 4·43-s + 8·47-s + 49-s + 10·53-s + 12·59-s + 2·61-s + 12·65-s − 12·67-s − 12·71-s + 14·73-s − 8·79-s − 12·83-s + 4·85-s + 2·89-s − 6·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.85·29-s + 1.43·31-s − 0.338·35-s + 0.986·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.56·59-s + 0.256·61-s + 1.48·65-s − 1.46·67-s − 1.42·71-s + 1.63·73-s − 0.900·79-s − 1.31·83-s + 0.433·85-s + 0.211·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8343407701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8343407701\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.48007357402525, −13.14999872117956, −12.43317994047913, −11.95155464083486, −11.66347265347458, −11.40876345283431, −10.61970698554239, −10.07607968770685, −9.737398234426699, −9.179789339827098, −8.562760013092630, −8.027449826354344, −7.570527569244620, −7.260839608627390, −6.772304480461548, −5.861668452069432, −5.512349319423703, −4.856397285660649, −4.288704804893600, −3.965176980189816, −3.207796911828762, −2.488932091315669, −2.094414836204572, −1.143595942826698, −0.2955898784972440,
0.2955898784972440, 1.143595942826698, 2.094414836204572, 2.488932091315669, 3.207796911828762, 3.965176980189816, 4.288704804893600, 4.856397285660649, 5.512349319423703, 5.861668452069432, 6.772304480461548, 7.260839608627390, 7.570527569244620, 8.027449826354344, 8.562760013092630, 9.179789339827098, 9.737398234426699, 10.07607968770685, 10.61970698554239, 11.40876345283431, 11.66347265347458, 11.95155464083486, 12.43317994047913, 13.14999872117956, 13.48007357402525
