| L(s) = 1 | + 3-s + 3·7-s − 2·9-s + 4·11-s − 3·13-s + 3·17-s + 3·21-s − 23-s − 5·25-s − 5·27-s + 29-s + 4·31-s + 4·33-s + 6·37-s − 3·39-s − 4·41-s + 4·47-s + 2·49-s + 3·51-s + 53-s + 9·59-s − 8·61-s − 6·63-s − 3·67-s − 69-s − 12·71-s − 3·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s − 2/3·9-s + 1.20·11-s − 0.832·13-s + 0.727·17-s + 0.654·21-s − 0.208·23-s − 25-s − 0.962·27-s + 0.185·29-s + 0.718·31-s + 0.696·33-s + 0.986·37-s − 0.480·39-s − 0.624·41-s + 0.583·47-s + 2/7·49-s + 0.420·51-s + 0.137·53-s + 1.17·59-s − 1.02·61-s − 0.755·63-s − 0.366·67-s − 0.120·69-s − 1.42·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.361205582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.361205582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.87906388968319, −13.62130325694251, −12.89040779258084, −12.14618148907859, −11.81102389111349, −11.56984644262231, −10.99956970506414, −10.27636193934710, −9.759342868764059, −9.408928902682996, −8.642367019808811, −8.446455864132304, −7.807763761415989, −7.437294636839618, −6.838602027071704, −6.011624178849984, −5.714534635268343, −5.025421488025779, −4.309011495152573, −4.061622934750754, −3.141289101739678, −2.726186253429562, −1.894528720254950, −1.477488806350506, −0.5553961312667402,
0.5553961312667402, 1.477488806350506, 1.894528720254950, 2.726186253429562, 3.141289101739678, 4.061622934750754, 4.309011495152573, 5.025421488025779, 5.714534635268343, 6.011624178849984, 6.838602027071704, 7.437294636839618, 7.807763761415989, 8.446455864132304, 8.642367019808811, 9.408928902682996, 9.759342868764059, 10.27636193934710, 10.99956970506414, 11.56984644262231, 11.81102389111349, 12.14618148907859, 12.89040779258084, 13.62130325694251, 13.87906388968319
