| L(s) = 1 | + 1.29·3-s + 0.772·5-s + 1.80·7-s − 1.32·9-s + 1.47·11-s − 2.67·13-s + 15-s − 4.96·17-s − 1.37·19-s + 2.33·21-s − 5.16·23-s − 4.40·25-s − 5.59·27-s − 8.75·29-s + 0.687·31-s + 1.91·33-s + 1.39·35-s + 3.87·37-s − 3.46·39-s + 2.74·41-s − 11.3·43-s − 1.02·45-s − 3.65·47-s − 3.75·49-s − 6.42·51-s − 1.37·53-s + 1.14·55-s + ⋯ |
| L(s) = 1 | + 0.747·3-s + 0.345·5-s + 0.680·7-s − 0.441·9-s + 0.445·11-s − 0.742·13-s + 0.258·15-s − 1.20·17-s − 0.314·19-s + 0.508·21-s − 1.07·23-s − 0.880·25-s − 1.07·27-s − 1.62·29-s + 0.123·31-s + 0.333·33-s + 0.234·35-s + 0.637·37-s − 0.555·39-s + 0.428·41-s − 1.72·43-s − 0.152·45-s − 0.533·47-s − 0.536·49-s − 0.899·51-s − 0.189·53-s + 0.153·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.29T + 3T^{2} \) |
| 5 | \( 1 - 0.772T + 5T^{2} \) |
| 7 | \( 1 - 1.80T + 7T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 2.67T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 + 5.16T + 23T^{2} \) |
| 29 | \( 1 + 8.75T + 29T^{2} \) |
| 31 | \( 1 - 0.687T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 + 3.65T + 47T^{2} \) |
| 53 | \( 1 + 1.37T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 + 2.28T + 61T^{2} \) |
| 67 | \( 1 - 8.95T + 67T^{2} \) |
| 71 | \( 1 + 4.42T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 9.80T + 89T^{2} \) |
| 97 | \( 1 + 3.10T + 97T^{2} \) |
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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
−8.080936395111150599003169722677, −7.59291340528236669209902630827, −6.59568679919216429437009869314, −5.89454142840686474620593258291, −5.02583980592419738013725831864, −4.19874751532140630879447366421, −3.40746323346185897396215724909, −2.20916298962390257648505168281, −1.89058796809487042177090097611, 0,
1.89058796809487042177090097611, 2.20916298962390257648505168281, 3.40746323346185897396215724909, 4.19874751532140630879447366421, 5.02583980592419738013725831864, 5.89454142840686474620593258291, 6.59568679919216429437009869314, 7.59291340528236669209902630827, 8.080936395111150599003169722677
