| L(s) = 1 | + 1.19·3-s − 2.70·5-s − 1.57·9-s − 3.94·11-s − 4.74·13-s − 3.23·15-s − 6.61·17-s − 19-s + 2.74·23-s + 2.33·25-s − 5.46·27-s − 4.65·29-s − 4.61·31-s − 4.70·33-s + 3.67·37-s − 5.67·39-s + 8.46·41-s + 3.01·43-s + 4.26·45-s − 11.2·47-s − 7.89·51-s + 12.4·53-s + 10.6·55-s − 1.19·57-s + 8.52·59-s − 10.1·61-s + 12.8·65-s + ⋯ |
| L(s) = 1 | + 0.689·3-s − 1.21·5-s − 0.524·9-s − 1.18·11-s − 1.31·13-s − 0.835·15-s − 1.60·17-s − 0.229·19-s + 0.573·23-s + 0.467·25-s − 1.05·27-s − 0.864·29-s − 0.828·31-s − 0.819·33-s + 0.604·37-s − 0.908·39-s + 1.32·41-s + 0.459·43-s + 0.635·45-s − 1.63·47-s − 1.10·51-s + 1.70·53-s + 1.44·55-s − 0.158·57-s + 1.10·59-s − 1.30·61-s + 1.59·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5163259199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5163259199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 2.70T + 5T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 23 | \( 1 - 2.74T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 - 8.46T + 41T^{2} \) |
| 43 | \( 1 - 3.01T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 8.52T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 + 1.95T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 1.07T + 79T^{2} \) |
| 83 | \( 1 + 0.253T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 0.369T + 97T^{2} \) |
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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
−7.82910558537061095030828473717, −7.45686103870885643449332973276, −6.76599385474872193460200618685, −5.67558855041384656775273189219, −4.96038646282560417917158846759, −4.25487154835124259586713511024, −3.52606627081198766051451725496, −2.59031749811330454904033381256, −2.22370400705704473517230259968, −0.31802715297014847106189025813,
0.31802715297014847106189025813, 2.22370400705704473517230259968, 2.59031749811330454904033381256, 3.52606627081198766051451725496, 4.25487154835124259586713511024, 4.96038646282560417917158846759, 5.67558855041384656775273189219, 6.76599385474872193460200618685, 7.45686103870885643449332973276, 7.82910558537061095030828473717
