| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s + 4·11-s − 12-s − 2·13-s − 4·14-s + 15-s + 16-s + 18-s − 19-s − 20-s + 4·21-s + 4·22-s + 8·23-s − 24-s + 25-s − 2·26-s − 27-s − 4·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.872·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.911395012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.911395012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.10792594765890, −12.79782858158366, −12.32786957296379, −12.02305938896263, −11.29511847801279, −11.06638902217073, −10.63060999415732, −9.782537174618606, −9.486880605932194, −9.156913537351374, −8.500313201006696, −7.620510295233326, −7.201198366197974, −6.861409274442398, −6.408360047829338, −5.795494175776664, −5.527770396490855, −4.704595241051229, −4.183992369105346, −3.796590759201725, −3.235660157344344, −2.667525234190612, −2.005050165197873, −0.9744618946809383, −0.5523568627657986,
0.5523568627657986, 0.9744618946809383, 2.005050165197873, 2.667525234190612, 3.235660157344344, 3.796590759201725, 4.183992369105346, 4.704595241051229, 5.527770396490855, 5.795494175776664, 6.408360047829338, 6.861409274442398, 7.201198366197974, 7.620510295233326, 8.500313201006696, 9.156913537351374, 9.486880605932194, 9.782537174618606, 10.63060999415732, 11.06638902217073, 11.29511847801279, 12.02305938896263, 12.32786957296379, 12.79782858158366, 13.10792594765890
