| L(s) = 1 | + 1.53·2-s + 3-s + 0.368·4-s − 3.54·5-s + 1.53·6-s − 4.90·7-s − 2.51·8-s + 9-s − 5.45·10-s + 0.368·12-s − 0.396·13-s − 7.55·14-s − 3.54·15-s − 4.60·16-s − 6.73·17-s + 1.53·18-s + 19-s − 1.30·20-s − 4.90·21-s − 4.65·23-s − 2.51·24-s + 7.54·25-s − 0.610·26-s + 27-s − 1.80·28-s + 1.12·29-s − 5.45·30-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.577·3-s + 0.184·4-s − 1.58·5-s + 0.628·6-s − 1.85·7-s − 0.887·8-s + 0.333·9-s − 1.72·10-s + 0.106·12-s − 0.109·13-s − 2.01·14-s − 0.914·15-s − 1.15·16-s − 1.63·17-s + 0.362·18-s + 0.229·19-s − 0.291·20-s − 1.07·21-s − 0.970·23-s − 0.512·24-s + 1.50·25-s − 0.119·26-s + 0.192·27-s − 0.341·28-s + 0.208·29-s − 0.995·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6137761406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6137761406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 13 | \( 1 + 0.396T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 23 | \( 1 + 4.65T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 - 8.55T + 37T^{2} \) |
| 41 | \( 1 + 2.06T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 0.222T + 53T^{2} \) |
| 59 | \( 1 + 5.80T + 59T^{2} \) |
| 61 | \( 1 + 9.14T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 - 7.37T + 71T^{2} \) |
| 73 | \( 1 + 0.284T + 73T^{2} \) |
| 79 | \( 1 - 7.91T + 79T^{2} \) |
| 83 | \( 1 + 0.796T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.84686824315499507875533249743, −7.12855901597454843204145850724, −6.51131448082832667284809416103, −5.94290946670190545780483156970, −4.76141497943038853008931456162, −4.19690467144038128869545645776, −3.61119169527594162710967138780, −3.16274233588436793881911217177, −2.34975239584398217241275523058, −0.30321238115773901659210871735,
0.30321238115773901659210871735, 2.34975239584398217241275523058, 3.16274233588436793881911217177, 3.61119169527594162710967138780, 4.19690467144038128869545645776, 4.76141497943038853008931456162, 5.94290946670190545780483156970, 6.51131448082832667284809416103, 7.12855901597454843204145850724, 7.84686824315499507875533249743
