L(s) = 1 | − 0.414·3-s − 5-s − 2.82·9-s − 11-s + 2.41·13-s + 0.414·15-s − 0.414·17-s + 2·19-s − 2.24·23-s + 25-s + 2.41·27-s + 29-s + 1.75·31-s + 0.414·33-s − 7.89·37-s − 0.999·39-s + 7.41·41-s − 0.343·43-s + 2.82·45-s + 10.4·47-s + 0.171·51-s + 9.41·53-s + 55-s − 0.828·57-s + 10.2·59-s − 1.17·61-s − 2.41·65-s + ⋯ |
L(s) = 1 | − 0.239·3-s − 0.447·5-s − 0.942·9-s − 0.301·11-s + 0.669·13-s + 0.106·15-s − 0.100·17-s + 0.458·19-s − 0.467·23-s + 0.200·25-s + 0.464·27-s + 0.185·29-s + 0.315·31-s + 0.0721·33-s − 1.29·37-s − 0.160·39-s + 1.15·41-s − 0.0523·43-s + 0.421·45-s + 1.51·47-s + 0.0240·51-s + 1.29·53-s + 0.134·55-s − 0.109·57-s + 1.33·59-s − 0.150·61-s − 0.299·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.221082793\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.221082793\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.414T + 3T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 0.414T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.24T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 + 7.89T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.343T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 9.41T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 1.17T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 1.65T + 89T^{2} \) |
| 97 | \( 1 + 0.0710T + 97T^{2} \) |
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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
−8.963412961604668777428915129694, −8.481316037666478387230815955718, −7.66495312428401522333430564742, −6.82615901724319307087658792645, −5.88753114520544006543845521613, −5.30005834957220383236982175395, −4.20127415637859527559155378124, −3.33574438539015703645461464277, −2.32123375246368362050578876699, −0.73737043072532815240532041581,
0.73737043072532815240532041581, 2.32123375246368362050578876699, 3.33574438539015703645461464277, 4.20127415637859527559155378124, 5.30005834957220383236982175395, 5.88753114520544006543845521613, 6.82615901724319307087658792645, 7.66495312428401522333430564742, 8.481316037666478387230815955718, 8.963412961604668777428915129694