| L(s) = 1 | − 2·5-s + 4·11-s − 13-s − 4·19-s − 6·23-s − 25-s + 5·31-s + 37-s + 4·41-s − 43-s − 4·47-s + 6·53-s − 8·55-s − 3·61-s + 2·65-s + 11·67-s + 14·71-s − 14·73-s + 13·79-s − 14·83-s − 6·89-s + 8·95-s + 9·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s − 0.277·13-s − 0.917·19-s − 1.25·23-s − 1/5·25-s + 0.898·31-s + 0.164·37-s + 0.624·41-s − 0.152·43-s − 0.583·47-s + 0.824·53-s − 1.07·55-s − 0.384·61-s + 0.248·65-s + 1.34·67-s + 1.66·71-s − 1.63·73-s + 1.46·79-s − 1.53·83-s − 0.635·89-s + 0.820·95-s + 0.913·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.378437824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378437824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−16.49949989400133, −15.98626855955607, −15.41164193409115, −14.86969123412271, −14.33642283798891, −13.80788316920060, −13.06891475935845, −12.28399837654499, −12.02223140919453, −11.40529035457726, −10.88217841886309, −10.01110965400595, −9.593146691258397, −8.746964306818604, −8.231752382424955, −7.692094136406808, −6.880084780918383, −6.374485565440550, −5.672391917896146, −4.639727514565110, −4.092005753594561, −3.628790447577065, −2.569188481489749, −1.687712661127974, −0.5496115720408969,
0.5496115720408969, 1.687712661127974, 2.569188481489749, 3.628790447577065, 4.092005753594561, 4.639727514565110, 5.672391917896146, 6.374485565440550, 6.880084780918383, 7.692094136406808, 8.231752382424955, 8.746964306818604, 9.593146691258397, 10.01110965400595, 10.88217841886309, 11.40529035457726, 12.02223140919453, 12.28399837654499, 13.06891475935845, 13.80788316920060, 14.33642283798891, 14.86969123412271, 15.41164193409115, 15.98626855955607, 16.49949989400133
