| L(s) = 1 | − 1.33·2-s − 1.95·3-s − 0.224·4-s − 2.95·5-s + 2.60·6-s + 3.88·7-s + 2.96·8-s + 0.827·9-s + 3.93·10-s + 0.439·12-s + 2.81·13-s − 5.18·14-s + 5.78·15-s − 3.50·16-s − 2.46·17-s − 1.10·18-s − 6.39·19-s + 0.664·20-s − 7.60·21-s + 6.21·23-s − 5.79·24-s + 3.73·25-s − 3.74·26-s + 4.25·27-s − 0.873·28-s + 9.76·29-s − 7.70·30-s + ⋯ |
| L(s) = 1 | − 0.942·2-s − 1.12·3-s − 0.112·4-s − 1.32·5-s + 1.06·6-s + 1.46·7-s + 1.04·8-s + 0.275·9-s + 1.24·10-s + 0.126·12-s + 0.779·13-s − 1.38·14-s + 1.49·15-s − 0.875·16-s − 0.596·17-s − 0.259·18-s − 1.46·19-s + 0.148·20-s − 1.66·21-s + 1.29·23-s − 1.18·24-s + 0.746·25-s − 0.734·26-s + 0.818·27-s − 0.165·28-s + 1.81·29-s − 1.40·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5954859019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5954859019\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.33T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 + 2.46T + 17T^{2} \) |
| 19 | \( 1 + 6.39T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 - 9.76T + 29T^{2} \) |
| 31 | \( 1 - 2.33T + 31T^{2} \) |
| 37 | \( 1 + 0.734T + 37T^{2} \) |
| 41 | \( 1 - 4.87T + 41T^{2} \) |
| 43 | \( 1 - 4.95T + 43T^{2} \) |
| 47 | \( 1 + 0.728T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 - 6.07T + 59T^{2} \) |
| 67 | \( 1 - 9.66T + 67T^{2} \) |
| 71 | \( 1 + 1.46T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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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
−8.180905200675637897389286147749, −7.28045137148227187728490843759, −6.75033397394015273433100052969, −5.81841356087512282453258631949, −4.80238916391126383989542718362, −4.58294844791465790807735012090, −3.88730760256893900949860621852, −2.45055995610596528317252491071, −1.18204998214144659484042233551, −0.58598349244601352503312282449,
0.58598349244601352503312282449, 1.18204998214144659484042233551, 2.45055995610596528317252491071, 3.88730760256893900949860621852, 4.58294844791465790807735012090, 4.80238916391126383989542718362, 5.81841356087512282453258631949, 6.75033397394015273433100052969, 7.28045137148227187728490843759, 8.180905200675637897389286147749
