| L(s) = 1 | + 2-s − 1.77·3-s + 4-s + 5-s − 1.77·6-s + 3.83·7-s + 8-s + 0.135·9-s + 10-s − 4.58·11-s − 1.77·12-s + 3.83·14-s − 1.77·15-s + 16-s + 3.86·17-s + 0.135·18-s + 5.15·19-s + 20-s − 6.79·21-s − 4.58·22-s + 4.98·23-s − 1.77·24-s + 25-s + 5.07·27-s + 3.83·28-s + 0.976·29-s − 1.77·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.02·3-s + 0.5·4-s + 0.447·5-s − 0.722·6-s + 1.45·7-s + 0.353·8-s + 0.0450·9-s + 0.316·10-s − 1.38·11-s − 0.511·12-s + 1.02·14-s − 0.457·15-s + 0.250·16-s + 0.938·17-s + 0.0318·18-s + 1.18·19-s + 0.223·20-s − 1.48·21-s − 0.976·22-s + 1.03·23-s − 0.361·24-s + 0.200·25-s + 0.976·27-s + 0.725·28-s + 0.181·29-s − 0.323·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.329303144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.329303144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 4.58T + 11T^{2} \) |
| 17 | \( 1 - 3.86T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 - 0.976T + 29T^{2} \) |
| 31 | \( 1 + 8.74T + 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 + 2.57T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 + 0.238T + 53T^{2} \) |
| 59 | \( 1 - 1.77T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 4.50T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 2.09T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 + 7.79T + 97T^{2} \) |
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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
−9.521615568137151903128873068873, −8.289628053330495637697821371848, −7.67680208321580041936402763848, −6.82476007230260929629181959581, −5.66735841397621179083979651804, −5.19287349525266252609497988731, −4.94319484323126611961376576574, −3.42753503656620918325297349161, −2.31392963544116640948404407758, −1.06222657876049792985641492257,
1.06222657876049792985641492257, 2.31392963544116640948404407758, 3.42753503656620918325297349161, 4.94319484323126611961376576574, 5.19287349525266252609497988731, 5.66735841397621179083979651804, 6.82476007230260929629181959581, 7.67680208321580041936402763848, 8.289628053330495637697821371848, 9.521615568137151903128873068873
