L(s) = 1 | + 3-s + 2·7-s + 9-s − 2·13-s − 3·17-s − 2·19-s + 2·21-s − 9·23-s + 27-s − 5·31-s − 2·37-s − 2·39-s + 3·41-s + 43-s + 3·47-s − 3·49-s − 3·51-s + 3·53-s − 2·57-s + 3·59-s + 2·61-s + 2·63-s + 5·67-s − 9·69-s − 6·71-s − 8·73-s + 79-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.727·17-s − 0.458·19-s + 0.436·21-s − 1.87·23-s + 0.192·27-s − 0.898·31-s − 0.328·37-s − 0.320·39-s + 0.468·41-s + 0.152·43-s + 0.437·47-s − 3/7·49-s − 0.420·51-s + 0.412·53-s − 0.264·57-s + 0.390·59-s + 0.256·61-s + 0.251·63-s + 0.610·67-s − 1.08·69-s − 0.712·71-s − 0.936·73-s + 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.198980440\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.198980440\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.36769396287101, −14.15584387483584, −13.56501952617620, −12.90523999756154, −12.57282586356319, −11.86279720217796, −11.46941188270324, −10.88921411816478, −10.22174264834601, −9.942876825429582, −9.145641684426087, −8.737616702272763, −8.231541039088403, −7.623695410536005, −7.302345398785767, −6.512874029174075, −5.954512279980299, −5.285361447722367, −4.614392864015068, −4.131531237130883, −3.584598426433577, −2.664826209098459, −2.068383964941782, −1.663042841029100, −0.4733739205485806,
0.4733739205485806, 1.663042841029100, 2.068383964941782, 2.664826209098459, 3.584598426433577, 4.131531237130883, 4.614392864015068, 5.285361447722367, 5.954512279980299, 6.512874029174075, 7.302345398785767, 7.623695410536005, 8.231541039088403, 8.737616702272763, 9.145641684426087, 9.942876825429582, 10.22174264834601, 10.88921411816478, 11.46941188270324, 11.86279720217796, 12.57282586356319, 12.90523999756154, 13.56501952617620, 14.15584387483584, 14.36769396287101