| L(s) = 1 | + 3.32·3-s + 2.28·5-s + 8.04·9-s − 2.44·11-s − 0.540·13-s + 7.58·15-s + 4.03·17-s + 19-s + 3.91·23-s + 0.209·25-s + 16.7·27-s + 5.85·29-s − 0.987·31-s − 8.11·33-s − 5.78·37-s − 1.79·39-s + 4.28·41-s − 6.87·43-s + 18.3·45-s + 1.82·47-s + 13.4·51-s + 7.01·53-s − 5.57·55-s + 3.32·57-s + 7.62·59-s − 14.7·61-s − 1.23·65-s + ⋯ |
| L(s) = 1 | + 1.91·3-s + 1.02·5-s + 2.68·9-s − 0.736·11-s − 0.150·13-s + 1.95·15-s + 0.979·17-s + 0.229·19-s + 0.816·23-s + 0.0419·25-s + 3.22·27-s + 1.08·29-s − 0.177·31-s − 1.41·33-s − 0.950·37-s − 0.287·39-s + 0.669·41-s − 1.04·43-s + 2.73·45-s + 0.266·47-s + 1.87·51-s + 0.963·53-s − 0.751·55-s + 0.440·57-s + 0.992·59-s − 1.88·61-s − 0.153·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.733579538\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.733579538\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.32T + 3T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 + 0.540T + 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 + 0.987T + 31T^{2} \) |
| 37 | \( 1 + 5.78T + 37T^{2} \) |
| 41 | \( 1 - 4.28T + 41T^{2} \) |
| 43 | \( 1 + 6.87T + 43T^{2} \) |
| 47 | \( 1 - 1.82T + 47T^{2} \) |
| 53 | \( 1 - 7.01T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 + 14.7T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 4.53T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 + 0.293T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.079720867253474674132225287688, −7.30725831441169917281193990171, −6.79959039019788105166022501960, −5.72780231853656479080432624887, −5.02420356033522909183948453692, −4.16518671614083493394664498217, −3.19352501455654213366940926580, −2.79228568627256662279113371259, −1.98171329952797981218208789196, −1.20982997037208918760424618417,
1.20982997037208918760424618417, 1.98171329952797981218208789196, 2.79228568627256662279113371259, 3.19352501455654213366940926580, 4.16518671614083493394664498217, 5.02420356033522909183948453692, 5.72780231853656479080432624887, 6.79959039019788105166022501960, 7.30725831441169917281193990171, 8.079720867253474674132225287688
