| L(s) = 1 | − 2·5-s − 3.12·11-s + 0.438·13-s − 5.12·17-s + 3.12·19-s − 23-s − 25-s − 3.56·29-s + 2.43·31-s + 8.24·37-s + 9.80·41-s + 8·43-s − 0.684·47-s − 7·49-s − 2·53-s + 6.24·55-s + 10.2·59-s − 4.24·61-s − 0.876·65-s − 3.12·67-s + 13.5·71-s − 14.6·73-s − 3.12·79-s + 14.2·83-s + 10.2·85-s − 11.3·89-s − 6.24·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.941·11-s + 0.121·13-s − 1.24·17-s + 0.716·19-s − 0.208·23-s − 0.200·25-s − 0.661·29-s + 0.437·31-s + 1.35·37-s + 1.53·41-s + 1.21·43-s − 0.0998·47-s − 49-s − 0.274·53-s + 0.842·55-s + 1.33·59-s − 0.543·61-s − 0.108·65-s − 0.381·67-s + 1.60·71-s − 1.71·73-s − 0.351·79-s + 1.56·83-s + 1.11·85-s − 1.20·89-s − 0.640·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120773823\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120773823\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 29 | \( 1 + 3.56T + 29T^{2} \) |
| 31 | \( 1 - 2.43T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 9.80T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 0.684T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 3.12T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 3.12T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.535185484525609968501016157686, −7.73580135228354232855793029670, −7.44229362665304822290993574684, −6.36900333951428284812280623387, −5.62550360975466260795004550715, −4.63465771887283410881463454096, −4.05740073099969839511991483442, −3.03936951224004166450175734061, −2.16336592544872534607461564469, −0.61040374913990718689447071714,
0.61040374913990718689447071714, 2.16336592544872534607461564469, 3.03936951224004166450175734061, 4.05740073099969839511991483442, 4.63465771887283410881463454096, 5.62550360975466260795004550715, 6.36900333951428284812280623387, 7.44229362665304822290993574684, 7.73580135228354232855793029670, 8.535185484525609968501016157686
