| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s − 15-s − 2·17-s − 19-s + 2·21-s + 2·23-s + 25-s − 27-s − 2·29-s − 2·35-s − 8·37-s − 6·41-s + 10·43-s + 45-s + 10·47-s − 3·49-s + 2·51-s + 8·53-s + 57-s + 4·59-s + 2·61-s − 2·63-s + 4·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.258·15-s − 0.485·17-s − 0.229·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.338·35-s − 1.31·37-s − 0.937·41-s + 1.52·43-s + 0.149·45-s + 1.45·47-s − 3/7·49-s + 0.280·51-s + 1.09·53-s + 0.132·57-s + 0.520·59-s + 0.256·61-s − 0.251·63-s + 0.488·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.02956637206014, −15.63695675633897, −15.06454168880357, −14.39848361200865, −13.71509255984484, −13.32319703494620, −12.70802317731547, −12.28707597839246, −11.65780775942024, −10.96641642698669, −10.48886829996151, −10.01231749038128, −9.271766142879187, −8.914184432891972, −8.144053314166000, −7.225959558251208, −6.856766220784965, −6.243067497422725, −5.587155488394908, −5.116772118605875, −4.218671237112946, −3.628727814416702, −2.722336122586438, −2.013173898278451, −0.9993884895165477, 0,
0.9993884895165477, 2.013173898278451, 2.722336122586438, 3.628727814416702, 4.218671237112946, 5.116772118605875, 5.587155488394908, 6.243067497422725, 6.856766220784965, 7.225959558251208, 8.144053314166000, 8.914184432891972, 9.271766142879187, 10.01231749038128, 10.48886829996151, 10.96641642698669, 11.65780775942024, 12.28707597839246, 12.70802317731547, 13.32319703494620, 13.71509255984484, 14.39848361200865, 15.06454168880357, 15.63695675633897, 16.02956637206014
