| L(s) = 1 | − 3-s + 9-s − 6·11-s + 13-s + 6·17-s − 5·19-s + 8·23-s − 27-s + 2·29-s + 4·31-s + 6·33-s − 37-s − 39-s + 6·41-s + 12·43-s − 12·47-s − 6·51-s − 6·53-s + 5·57-s − 8·59-s + 15·61-s + 5·67-s − 8·69-s − 6·71-s + 73-s − 79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 1.45·17-s − 1.14·19-s + 1.66·23-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 1.04·33-s − 0.164·37-s − 0.160·39-s + 0.937·41-s + 1.82·43-s − 1.75·47-s − 0.840·51-s − 0.824·53-s + 0.662·57-s − 1.04·59-s + 1.92·61-s + 0.610·67-s − 0.963·69-s − 0.712·71-s + 0.117·73-s − 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 15 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.01796702866006, −12.73197207966280, −12.35935021818565, −11.71307363801688, −11.16289949111959, −10.79530850009124, −10.49718381769475, −9.961998842049604, −9.562413243417075, −8.904046351890809, −8.298832404262073, −8.009643878223959, −7.415023957674834, −7.095751773989084, −6.262144996919622, −6.030329508470257, −5.411726557931841, −4.839663710977368, −4.738966416056504, −3.817574670504346, −3.247733737907125, −2.681142809802973, −2.239977506098656, −1.273523269938303, −0.7867149316054470, 0,
0.7867149316054470, 1.273523269938303, 2.239977506098656, 2.681142809802973, 3.247733737907125, 3.817574670504346, 4.738966416056504, 4.839663710977368, 5.411726557931841, 6.030329508470257, 6.262144996919622, 7.095751773989084, 7.415023957674834, 8.009643878223959, 8.298832404262073, 8.904046351890809, 9.562413243417075, 9.961998842049604, 10.49718381769475, 10.79530850009124, 11.16289949111959, 11.71307363801688, 12.35935021818565, 12.73197207966280, 13.01796702866006
