| L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s + 6·11-s − 2·13-s − 2·15-s + 6·17-s − 19-s + 4·21-s − 6·23-s − 25-s − 27-s + 4·29-s − 6·33-s − 8·35-s − 6·37-s + 2·39-s + 12·41-s + 2·45-s + 2·47-s + 9·49-s − 6·51-s + 12·53-s + 12·55-s + 57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.516·15-s + 1.45·17-s − 0.229·19-s + 0.872·21-s − 1.25·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s − 1.04·33-s − 1.35·35-s − 0.986·37-s + 0.320·39-s + 1.87·41-s + 0.298·45-s + 0.291·47-s + 9/7·49-s − 0.840·51-s + 1.64·53-s + 1.61·55-s + 0.132·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.520596231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.520596231\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.601247811672175714836348553087, −8.715226736035786393888765745508, −7.43193207148931676014482884977, −6.64778446909803345118823606838, −6.07519939375793332636005391714, −5.55235239605946024326730288350, −4.18067877179465198004011958335, −3.45391489076270786582601924875, −2.18255773381365998796372444819, −0.881283478455129418321128415443,
0.881283478455129418321128415443, 2.18255773381365998796372444819, 3.45391489076270786582601924875, 4.18067877179465198004011958335, 5.55235239605946024326730288350, 6.07519939375793332636005391714, 6.64778446909803345118823606838, 7.43193207148931676014482884977, 8.715226736035786393888765745508, 9.601247811672175714836348553087
