| L(s) = 1 | − 1.26·2-s − 9.40·3-s − 6.40·4-s − 0.325·5-s + 11.8·6-s − 8.80·7-s + 18.1·8-s + 61.3·9-s + 0.410·10-s − 23.6·11-s + 60.1·12-s − 23.2·13-s + 11.1·14-s + 3.05·15-s + 28.2·16-s − 125.·17-s − 77.5·18-s − 16.8·19-s + 2.08·20-s + 82.7·21-s + 29.9·22-s + 127.·23-s − 171.·24-s − 124.·25-s + 29.3·26-s − 322.·27-s + 56.3·28-s + ⋯ |
| L(s) = 1 | − 0.446·2-s − 1.80·3-s − 0.800·4-s − 0.0290·5-s + 0.808·6-s − 0.475·7-s + 0.804·8-s + 2.27·9-s + 0.0129·10-s − 0.648·11-s + 1.44·12-s − 0.496·13-s + 0.212·14-s + 0.0526·15-s + 0.441·16-s − 1.79·17-s − 1.01·18-s − 0.203·19-s + 0.0232·20-s + 0.860·21-s + 0.289·22-s + 1.15·23-s − 1.45·24-s − 0.999·25-s + 0.221·26-s − 2.30·27-s + 0.380·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.26T + 8T^{2} \) |
| 3 | \( 1 + 9.40T + 27T^{2} \) |
| 5 | \( 1 + 0.325T + 125T^{2} \) |
| 7 | \( 1 + 8.80T + 343T^{2} \) |
| 11 | \( 1 + 23.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 16.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 127.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 303.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 245.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 185.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 1.86T + 1.03e5T^{2} \) |
| 53 | \( 1 - 261.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 417.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 699.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 279.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 675.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 732.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 200.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−8.655974115401117536905915124538, −7.51073888753211119168387702939, −6.86011598439455820381766184301, −6.07723459137498591222561630147, −5.08946495603802623140250087850, −4.76120812625282337138215126697, −3.74548529317092745453985563978, −2.04335191575242413818340801297, −0.67073495555897230798352543142, 0,
0.67073495555897230798352543142, 2.04335191575242413818340801297, 3.74548529317092745453985563978, 4.76120812625282337138215126697, 5.08946495603802623140250087850, 6.07723459137498591222561630147, 6.86011598439455820381766184301, 7.51073888753211119168387702939, 8.655974115401117536905915124538
