| L(s) = 1 | − 2.41·2-s − 3-s + 3.82·4-s + 5-s + 2.41·6-s − 4.41·8-s + 9-s − 2.41·10-s − 2·11-s − 3.82·12-s + 1.58·13-s − 15-s + 2.99·16-s + 6.82·17-s − 2.41·18-s − 2.82·19-s + 3.82·20-s + 4.82·22-s + 5.24·23-s + 4.41·24-s − 4·25-s − 3.82·26-s − 27-s − 10.4·29-s + 2.41·30-s + 3.65·31-s + 1.58·32-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 0.577·3-s + 1.91·4-s + 0.447·5-s + 0.985·6-s − 1.56·8-s + 0.333·9-s − 0.763·10-s − 0.603·11-s − 1.10·12-s + 0.439·13-s − 0.258·15-s + 0.749·16-s + 1.65·17-s − 0.569·18-s − 0.648·19-s + 0.856·20-s + 1.02·22-s + 1.09·23-s + 0.901·24-s − 0.800·25-s − 0.750·26-s − 0.192·27-s − 1.93·29-s + 0.440·30-s + 0.656·31-s + 0.280·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 3.65T + 31T^{2} \) |
| 37 | \( 1 + 7.82T + 37T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 + 0.414T + 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 + 5.82T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 3.17T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 - 2.89T + 97T^{2} \) |
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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
−7.75275145995101495903977584035, −7.29804932027584102211462132163, −6.47993232990247850815679307052, −5.74650159126184745197119887211, −5.20005055125239507742300537575, −3.91553066474489064364651345319, −2.87427296301054287901307470610, −1.86889777355681597143620743459, −1.12543954085578698689542637075, 0,
1.12543954085578698689542637075, 1.86889777355681597143620743459, 2.87427296301054287901307470610, 3.91553066474489064364651345319, 5.20005055125239507742300537575, 5.74650159126184745197119887211, 6.47993232990247850815679307052, 7.29804932027584102211462132163, 7.75275145995101495903977584035
