| L(s) = 1 | + 2-s − 2.76·3-s + 4-s + 1.76·5-s − 2.76·6-s − 3.62·7-s + 8-s + 4.62·9-s + 1.76·10-s − 2.76·12-s + 2.62·13-s − 3.62·14-s − 4.86·15-s + 16-s − 4.62·17-s + 4.62·18-s − 19-s + 1.76·20-s + 10.0·21-s + 3.62·23-s − 2.76·24-s − 1.89·25-s + 2.62·26-s − 4.49·27-s − 3.62·28-s + 5.38·29-s − 4.86·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.59·3-s + 0.5·4-s + 0.787·5-s − 1.12·6-s − 1.37·7-s + 0.353·8-s + 1.54·9-s + 0.557·10-s − 0.797·12-s + 0.728·13-s − 0.969·14-s − 1.25·15-s + 0.250·16-s − 1.12·17-s + 1.09·18-s − 0.229·19-s + 0.393·20-s + 2.18·21-s + 0.756·23-s − 0.563·24-s − 0.379·25-s + 0.515·26-s − 0.864·27-s − 0.685·28-s + 1.00·29-s − 0.888·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.76T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 7 | \( 1 + 3.62T + 7T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 17 | \( 1 + 4.62T + 17T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 + 2.61T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 + 5.11T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 4.59T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 0.864T + 71T^{2} \) |
| 73 | \( 1 - 5.79T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 1.10T + 97T^{2} \) |
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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
−7.58780688345613118699659718382, −6.65335813309611636240154420846, −6.30417001790904337294603493733, −5.98566139788073694277556462028, −5.09580002707600189041794908821, −4.43605384307644530211188196081, −3.48030523334629787585003376169, −2.49701711623085969963799812444, −1.28726585956882012974288788040, 0,
1.28726585956882012974288788040, 2.49701711623085969963799812444, 3.48030523334629787585003376169, 4.43605384307644530211188196081, 5.09580002707600189041794908821, 5.98566139788073694277556462028, 6.30417001790904337294603493733, 6.65335813309611636240154420846, 7.58780688345613118699659718382
