| L(s) = 1 | + 4·3-s + 2·5-s − 24·7-s − 11·9-s + 44·11-s + 22·13-s + 8·15-s + 44·19-s − 96·21-s + 56·23-s − 121·25-s − 152·27-s − 198·29-s + 160·31-s + 176·33-s − 48·35-s + 162·37-s + 88·39-s + 198·41-s + 52·43-s − 22·45-s + 528·47-s + 233·49-s − 242·53-s + 88·55-s + 176·57-s − 668·59-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.178·5-s − 1.29·7-s − 0.407·9-s + 1.20·11-s + 0.469·13-s + 0.137·15-s + 0.531·19-s − 0.997·21-s + 0.507·23-s − 0.967·25-s − 1.08·27-s − 1.26·29-s + 0.926·31-s + 0.928·33-s − 0.231·35-s + 0.719·37-s + 0.361·39-s + 0.754·41-s + 0.184·43-s − 0.0728·45-s + 1.63·47-s + 0.679·49-s − 0.627·53-s + 0.215·55-s + 0.408·57-s − 1.47·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.520201629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520201629\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 198 T + p^{3} T^{2} \) |
| 31 | \( 1 - 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 - 198 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 242 T + p^{3} T^{2} \) |
| 59 | \( 1 + 668 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 728 T + p^{3} T^{2} \) |
| 73 | \( 1 + 154 T + p^{3} T^{2} \) |
| 79 | \( 1 - 656 T + p^{3} T^{2} \) |
| 83 | \( 1 - 236 T + p^{3} T^{2} \) |
| 89 | \( 1 - 714 T + p^{3} T^{2} \) |
| 97 | \( 1 - 478 T + p^{3} T^{2} \) |
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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
−8.967064107528420254201168142099, −7.87265269892346013155601616796, −7.19253060597324903211753360867, −6.09415485384843346630046400759, −5.93330512584598124963096275123, −4.40315153166926754568450556174, −3.53388325770876880467029166150, −3.01679767170346679095958223456, −1.92231191999183018318929843871, −0.67305611067297595975822037289,
0.67305611067297595975822037289, 1.92231191999183018318929843871, 3.01679767170346679095958223456, 3.53388325770876880467029166150, 4.40315153166926754568450556174, 5.93330512584598124963096275123, 6.09415485384843346630046400759, 7.19253060597324903211753360867, 7.87265269892346013155601616796, 8.967064107528420254201168142099
